UC-NRLF 








IN MEMORIAM 
FLOR1AN CAJORI 





AV^C, 



HIGHER 



AEITHMETIC 



BY 



WOOSTER WOODRUFF BEMAN 

H 

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN 
AND 

DAVID EUGENE SMITH 

PROFESSOR OF MATHEMATICS IN THE MICHIGAN STATE 
NORMAL COLLEGE 



BOSTON, U.S.A., AND LONDON 
GINN & COMPANY, PUBLISHERS 



1897 



COPYRIGHT, 1897, BY 
WOOSTEB WOODRUFF BEMAN AND DAVID EUGENE SMITH 



ALL RIGHTS RESERVED 



CAJOW 



PREFACE. 



1. THE present work has been prepared with the belief 
that it will be of service to progressive teachers in American 
high schools, academies, and normal schools. As indicated 
by its title, it is intended for those who are taking up the 
subject a second time with the desire to review and extend 
the knowledge previously acquired. The purpose of the 
authors is more fully set forth in the following statement 
of some of the distinctive features of the book. 

2. The applied problems refer to the ordinary commer- 
cial life of to-day, or they deal with elementary questions 
arising in the laboratory, or they are inserted for general 
information. The fact that tradition has furnished the 
schools with a mass of inherited puzzles which give a false 
notion of business, that in an age of science and invention 
these subjects have found no place in the arithmetics, and 
that the common graphic methods of representing statistics 
are not seen in the schools, has not deterred the authors 
from attempting to modernize the subject. At the same 
time they believe that the exercises will be found much 
more straightforward and simple than those with which 
the average text-book has so long been encumbered. 

3. Problems in pure arithmetic in the high school are 
intended to furnish training in mathematical analysis. This 
is almost their only justification. Hence, the attempt has 
been made to lead the pupil to a clear understanding of 

911349 



IV PREFACE. 

such subjects as the greatest common divisor, the multi- 
plication and division of fractions, the square and cube 
roots, etc. To this end it has been necessary to resort 
to the literal notation. Most students will know enough 
algebra for this purpose, but for those who do not the 
necessary foundation can be laid in two or three lessons. 

4. The work being intended as a review, it has not been 
thought necessary to attempt a definition of every arith- 
metical term which is employed. A table of the most 
common terms is given on p. vii, with definitions and 
etymologies. 

5. The work being intended only for those teachers who 
recognize in arithmetic an instrument for mental training, 
no rules are given. 

6. .Teachers are urged to follow the suggestions laid 
down in the work, with respect to the omission of such 
chapters or portions of chapters as are not adapted to their 
pupils, and to change the sequence to suit their own tastes. 
It is hardly necessary to suggest that only a portion of 
the exercises should be attempted by any one class, the 
advantage of changing from term to term being evident. 

7. While the authors have not failed to consult the 
leading French, German, Italian, and English arithmetics, 
they have taken but few problems from these sources. To 
Day's " Electric Light Arithmetic " (London, 1887) they 
are, however, indebted for several exercises. 

8. In the work in mensuration several figures have been 
taken from the authors' "Plane and Solid Geometry" 
(Boston, Ginn & Co.). For the scientific treatment of that 
subject and for additional exercises, teachers are referred 
to the text-book mentioned. 

W. W. BEMAN. 
D. E. SMITH. 
JULY 1, 1897. 



TABLE OF CONTENTS. 

PAGE 

PREFACE iii 

TABLE OF CONTENTS ........ v 

DEFINITIONS AND ETYMOLOGIES ...... vii 

SYMBOLS AND ABBREVIATIONS ...... xvii 

CHAPTER I. NOTATION AND THE FUNDAMENTAL OPERATIONS 1 

I. WRITING AND READING NUMBERS .... 1 

II. CHECKS ......... 3 

III. ADDITION ......... 4 

IV. SUBTRACTION ........ 5 

V. MULTIPLICATION 6 

VI. DIVISION 10 

AXIOMS ......... 13 

FUNDAMENTAL LAWS ...... 14 

CHAPTER II. FACTORS AND MULTIPLES .... 15 

I. TESTS OF DIVISIBILITY . . . . . . 15 

II. CASTING OUT NINES ....... 19 

III. GREATEST COMMON DIVISOR ..... 21 

IV. LEAST COMMON MULTIPLE ...... 24 

CHAPTER III. COMMON FRACTIONS . . . 26 

CHAPTER IV. POWERS AND ROOTS 32 

I. INVOLUTION ........ 32 

II. SQUARE ROOT 34 

III. CUBE ROOT 39 

CHAPTER V. THE FORMAL SOLUTION OF PROBLEMS. . 41 

I. SYMBOLS 42 

II. LANGUAGE ......... 44 

III. METHODS 44 

IV. CHECKS 49 

CHAPTER VI. MEASURES 51 

I. COMPOUND NUMBERS ....... 54 

II. THE METRIC SYSTEM 59 

III. MEASURES OF TEMPERATURE 64 



VI TABLE OF CONTENTS. 

CHAPTER VII. MENSURATION 66 

CHAPTER VIII. LONGITUDE AND TIME . . . . 80 

CHAPTER IX. RATIO AND PROPORTION .... 87 

I. RATIO 87 

II. PROPORTION 96 

CHAPTER X. SERIES 104 

I. ARITHMETIC SERIES 105 

II. GEOMETRIC SERIES 107 

CHAPTER XI. LOGARITHMS ...... 110 

CHAPTER XII. GRAPHIC ARITHMETIC . . . .122 

CHAPTER XIII. INTRODUCTION TO PERCENTAGE . . 126 

CHAPTER XIV. COMMERCIAL DISCOUNTS AND PROFITS . 132 

CHAPTER XV. INTEREST, PROMISSORY NOTES, PARTIAL 

PAYMENTS 136 

I. SIMPLE INTEREST ....... 136 

II. PROMISSORY NOTES ....... 141 

III. PARTIAL PAYMENTS 143 

IV. COMPOUND INTEREST . . . . . 145 
V. ANNUAL INTEREST 147 

CHAPTER XVI. BANKING BUSINESS .... 148 

I. DEPOSITS AND CHECKS ...... 148 

II. LENDING MONEY 150 

III. DISCOUNTING NOTES ....... 152 

CHAPTER XVII. EXCHANGE 154 

CHAPTER XVIII. GOVERNMENT REVENUES . . .163 

I. THE UNITED STATES GOVERNMENT .... 163 

II. STATE AND LOCAL TAXES 165 

CHAPTER XIX. COMMISSION AND BROKERAGE . . 168 

CHAPTER XX. STOCKS AND BONDS 171 

CHAPTER XXI. INSURANCE 177 

CHAPTER XXII. MISCELLANEOUS EXERCISES. . . .179 

APPENDIX NOTE I . 189 

II 190 

III 192 



DEFINITIONS AND ETYMOLOGIES. 



THE following list includes such definitions as are apt to be needed 
for reference, together with pronunciations and etymologies. The 
latter are those given by the Century Dictionary. 

KEY. L. Latin, G. Greek, F. French, ML. Mediaeval Latin, LL. 
Low Latin, AS. Anglo-Saxon, ME. Middle English, dim. diminutive, 
fern, feminine. 

a, fat, & fate, a'/ar, a, fall, a ask, a /are, 

e met, e mete, e her, i pin, I pine, o not, 

6 note, o move, 6 nor, u tub, u mute, u. pull. 

s as in leisure. 

A single dot under a vowel indicates its abbreviation. 
A double dot under a vowel indicates that the vowel approaches the 
short sound of u, as in put. 

The numerals refer to pages in the arithmetic. 

Abscissa (ab-sis'a). L. cut off. A certain line used in determining 
the position of a point in a plane. 68. 

Abstract (ab'strakt). L. abstractus, drawn away. An abstract num- 
ber is a number not designated as referring to any particular class 
of objects. E.g., 7, as distinguished from 7 ft. which is a concrete 
number. 

Addition (a-dish'on). L. addere, to increase. The uniting of two or 
more numbers in one sum. 

Ampere (am-par'). Term adopted by the Electric Congress at Paris, 
1881. Name of French electrician, Andre' Marie Ampere, d. 1836. 
The unit employed in measuring the strength of an electric cur- 
rent. 101. 

Analysis (a-nal'i-sis). G. a loosing, a resolution of a whole into its 
parts. A form of reasoning from a whole to its parts. 

Antecedent (an-te-se'-dent). L. antecedere, to go before. The first 
of two terms of a ratio. 

Antilogarithm (an-ti-logVrithm). G. anti-, opposite to, + logarithm. 
The number corresponding to a logarithm. 117. 



Vlll HIGHER ARITHMETIC. 

Approximation (a-prok-si-ma'shpn). L. ad", to, + proximare, to come 
near, (a) A continual approach to a true result, (b) A result so 
near the truth as to be sufficient for a given purpose. 

Are (ar or ar). L. area, a piece of level ground. A square deka- 
ineter, or 119.6 sq. yds. 

Area (a're-a). See Are. The superficial contents of any figure or 
surface. 

Arithmetic (a-rith'me-tik ; as adj., ar-ith-inet'ik). G. arithmos, num- 
ber. The theory of numbers, the art of computation, and the 
applications of numbers to science and business. 

Arithmetic series. See Series. 101. 

Associative law (a-so'shi-a-tiv). L. ad 1 , to, + sociare, to join. The 
law which states that certain operations give the same result 
whether they first unite two quantities A and B, and then unite 
the result to a third quantity C ; or first unite B and C, and then 
unite the result to J., the order of the quantities being preserved. 

Average (av'e-raj). Etymology obscure. The result of adding several 
quantities and dividing the sum by the number of quantities. 

Avoirdupois (av"or-du-poiz'). F. aver, goods, + de, of, + jpois, weight. 
A system of weight in which 1 Ib. = 16 oz. = approximately 7000 
troy grains. 

Axiom (ak'si-om). G. axioma, a requisite, a self-evident principle. 
A simple statement, of a general nature, so obvious that its truth 
may be taken for granted. 

Bank discount (bangk). ML. bancus, bench. See Discount. 

Base (bas). LL. bassus, low. (a) The line or surface forming that 
part of a figure on which it is supposed to stand, (b) The base of 
a system of logarithms is the number which, raised to the power 
indicated by the logarithm, gives the number to which the loga- 
rithm belongs, (c) In percentage, the number which is multiplied 
by the rate to produce the percentage. 

Bond (bond). AS. bindan, to bind. An obligation, under seal, to 
pay money. It may be issued by a government, a railway corpora- 
tion, a private individual, etc. 

Broker (bro'ker). Originally, one who manages. An agent. 

Brokerage (bro'ker-aj). The fee or commission given to a broker. 

Cancel (kan'sel). L. cancelli, a lattice. Originally, to draw lines 
across a calculation. To strike out or eliminate as a common 
factor in the terms of a fraction, a common term in the two mem- 
bers of an equation, etc. 



DEFINITIONS AND ETYMOLOGIES. IX 

Cast out nines. 14. 

Characteristic (kar^ak-te-ris'tik). G. characterizein, to designate. 112. 

Check. ME. cheker, a chess board. To verify ; that is, to mark off 

as having been examined. 
Circle (sir'kl). L. circulus, dim. of circus (G. kirkos), a ring. A plane 

figure whose periphery is everywhere equally distant from a point 

within it, the center. 

Circulating decimal (ser'ku-lat-ing). 109. 
Circumference (ser-kum'fe-rens). L. circum, around (see Circle), + 

/erre, to bear. The line which bounds a circle. 
Cologarithm (ko-log'a-rithm). 118. 
Commission (kp-mish'pn). L. com-, together, + mittere, to send. The 

act of intrusting ; hence, a fee paid to one who is intrusted. 
Common denominator (kom'pn). L. communis, general, universal. 

A denominator common to two or more fractions. 
Common fraction. See Fraction. A fraction in which both terms 

are written out in full, as distinguished from a decimal fraction. 
Common multiple. See Multiple. A multiple of two or more expres- 
sions is a common multiple of those expressions. 
Commutative law (kp-mu'ta-tiv). L. com-, intensive, + mutare, to 

change. The law which states that the order in which elements 

are combined is indifferent. 
Complement (kom'ple-ment). L. com-, intensive, + plere, to fill. A 

number added to a. second number to complete a third. 
Complex fraction (kom'pleks). L. com-, together, + plectere, to 

weave. (See Fraction.) A common fraction whose numerator 

or denominator contains a common fraction. 
Composite number (kom-poz'it). L. com-, together, + ponere, to put. 

A. number which can be exactly divided by a number exceeding 

unity. 

Compound interest (kom'pound). See Composite. 145. 
Compound number. 54. 
Concrete number (kon'kret or kpn-kret'). L. concretus, grown together. 

A number which specifies the unit, as 3 ft. In the case of 3 ft. , 3 

is strictly the number and 1 ft. is the unit ; it is, however, conven- 
ient to designate 3 as an abstract number and 3 ft. as a concrete 

number. 
Cone (kon). G. fconos, a cone. The elementary form considered in 

arithmetic is a solid generated by the revolution of a right-angled 

triangle about one of its sides as an axis. 



X HIGHER ARITHMETIC. 

Consequent (kon'se-kwent). L. com-, together, + sequi, to follow. 
The latter of the two terms of a ratio. 

Consignee (kon-sl-ne'). L. com-, together, + signare, to seal. One 
who has the care or disposal of goods upon consignment. 

Consignor (kon-si'nor or kon-si-n6r'). A person who makes a con- 
signment. 

Corporation (kor-po-ra'shon). L. corporare, to form into a body, from 
corpus, a body. An artificial person created by law from a group 
of natural persons and having a continuous existence irrespective 
of that of its members. 

Coupon bond (ko'pon). F. couper, to cut. A bond, usually of a state 
or corporation, for the payment of money at a future day, with 
severable tickets or coupons annexed, each representing an instal- 
ment of interest, which may be conveniently cut off for collection 
as they fall due. 

Cube (kub). G. kubos, a die, a cube, (a) A regular solid with six 
square faces, (b) To raise to the third power, (c) The third power 
of a number. 

Cube root. The cube root of a perfect third power is one of the three 
equal factors of that power. A number which has not a perfect 
third power has not three equal factors. It is, however, said to 
have a cube root to any required degree of approximation. Thus, 
the cube root of n to 0. 1 is that number of tenths whose cube differs 
from n by less than the cube of any other number of tenths. 

Cylinder (sirin-der). G. kylindros, from kyliein, to roll. The ele- 
mentary form considered in arithmetic is a solid generated by the 
revolution of a rectangle about one of its sides as an axis. 

Date line (daf lin). The fixed boundary line between neighboring 
regions where the calendar day is different. 

Decimal (des'i-mal). L. decem, ten. Pertaining to ten. 

Denominator (de-nom'i-na-tor). L. denominare, to name. 27. 

Diagonal (dl-ag'o-nal). G. dia, through, + gonia, corner, angle. A 
line through the angles of a figure, but not lying in its sides or faces. 

Difference (dif'e-rens). L. differ ens, different. The difference between 
two numbers is the number which added to either will produce the 
other. In arithmetic it is usually taken as the number which 
added to the smaller will produce the larger. 

Digit (dij'it). L. digitus, finger. The number represented by any 

one of the ten symbols 0, 1, 2, 9. The term is more often used 

to designate one of the ten symbols mentioned. 



DEFINITIONS AND ETYMOLOGIES. xi 

Directly proportional. 97. 

Discount (dis'kount). L. dis, away, + computare, to count. An 

allowance or deduction made from the customary or normal price, 

or from a sum due or to be due at a future time. Bank discount 

' is simple interest paid in advance, reckoned on the amount of a 

note. 

Distributive law (dis-trib'u-tiv). 14. 

Dividend (div'i-dend). L. dividere, to divide, (a) A number or 
quantity to be divided by another called the divisor, (b) A divi- 
sion of profits to be distributed proportionately among stockholders. 

Divisibility (di-viz-i-bil'i-ty). The quality of being divisible without 
a remainder. Usually applied in speaking of abstract integers. 

Division (di-vizh'pn). 29. 

Divisor (di-vi'zor). See Dividend. 

Draft (draft). AS. dragan, to draw. A writing directing the pay- 
ment of money on account of the drawer. 155, 156. 

Drawee of a draft. One on whom the order is drawn. 

Drawer of a draft. One who draws the order for the payment of 
money. 

Duty (du'ti). Due + ty. Sum of money levied by a government 
upon goods imported from abroad. 

Equal (e'kwal). L. aequalis, equal. Having the same value. 

Equation (e-kwa'shpn). A proposition asserting the equality of two 
quantities and expressed by the sign = between them. In algebra, 
an equality which exists only for particular values of certain letters 
called the unknown quantities. 

Equilateral (e-kwi-lat'e-ral). L. aequus, equal, + latus, side. Hav- 
ing all the sides equal. 

Evolution (ev'o-lu'shon). L. evolvere, to unroll. The extraction of 
roots from powers. 

Exchange (eks-chanj'). ML. ex, out, + cambiare, to change. The 
transmission of the equivalent of money from one place to another, 
such equivalent being redeemable in the money of the place to 
which it is sent. 

Exponent (eks-po'nent). L. exponere, to set forth, indicate. A symbol 
placed above and at the right of another symbol (the base) to 
denote that the latter is to be raised to a power. Eor general 
meaning, see pages 32, 33. 

Extremes (eks-tremz'). L. extremus, outermost. The first and last 
terms of a proportion or of any other related series of terms. 



Xll HIGHER ARITHMETIC. 

Face (fas). L. fades, face. The principal sum due on a note, bond, 

policy, etc. 
Factor (fak'tor). L. facer e, to do. One of two or more numbers 

which when multiplied together produce a given number. 
Fraction (frak'shon). L. frangere, to break. 26. 
Fractional unit. One of the equal parts of unity. 
Geometric series (je-o-met'rik). G. geometria, geometry. 104. 
Grace (gras). L. grains, dear. 141. 
Gram (gram). 59. 
Greatest common divisor of two or more integers is the greatest 

integer which will divide each of them without a remainder. 
Hypotenuse (hi-pot'e-nus). G. hypo, under, + teinein, to stretch. 

The side of a right-angled triangle opposite the right angle. 
Improper fraction. A common fraction whose terms are positive, 

and whose numerator is not less than its denominator. 
Index notation. 1. 
Insurance (in-shor'ans). OF. enseurer, to insure. A contract by 

which one party for an agreed consideration undertakes to com- 
pensate another for loss on a specific thing. 
Integer (in'te-jer). L., a whole number. 
Interest (in'ter-est). L. interest, it concerns. A sum paid for the 

use of money. 

Inversely proportional (in-versli). 97. 
Involution (in-vo-lu'shon). L. involvere, to roll up. Multiplication 

of a quantity into itself any number of times. 
Isosceles (i-sos'e-lez). G. isos, equal, + skelos, leg. Having two sides 

equal. 

Least common denominator. The least denominator which can be 

common to two or more common fractions. 
Least common multiple. The least integer which is a multiple of two 

or more given integers. 
Lever (lev'er or leaver). L. levare, to raise. A bar acted upon at 

different points by two forces which severally tend to rotate it in 

opposite directions about a fixed point called the fulcrum. 
Liter (le'ter). G. litra, a pound. The unit of capacity in the metric 

system ; a cubic decimeter. 59. 
Logarithm (log'a-rithm). G. logos, word, 4- arithmos, number. The 

exponent of the power to which a number called the base (in the 

common system, 10) must be raised to produce a number. 



DEFINITIONS AND ETYMOLOGIES. Xlll 

Longitude (lon'ji-tud). L. longus, long. The angle at the pole be- 
tween two meridians, one of which, the Prime Meridian, passes 
through some conventional point and from which the angle is 
measured. 

Mantissa (man-tis'a). L., something left over. 112. 

Maturity (ma-tu'ri-ti), L. maturus, mature. The time when a note 

or bond becomes due. 

Means (menz). The second and third terms of a proportion. 
Measure (mezh'ur). L. mensura, measure, (a) A unit or standard 

adopted to determine the length, volume, or other quantity of some 

other object, (b) The determination of quantity by the use of a 

unit. 

Mensuration (men-su-ra'shon). The science of measuring. 
Meridian (me-rid'i-an). L. meridianus, belonging to mid-day. A 

semi-circumference passing through the poles. 
Meter (me'ter). G. metron, measure. Unit of length in the metric 

system. 59. 
Metric (met'rik). 59. 

Mikron (ml'kron). G. mikros, small. Millionth part of a meter. 
Minuend (min'u-end). L. minuere, to lessen. The number from 

which another number is subtracted. 
Mixed number. The sum of an integer and a fraction. 
Multiple (murti-pl). L. multus, many, + plus, akin to E. fold. A 

number produced by multiplying an integer by an integer. 
Multiplicand (murti-pli-kand). A number multiplied by another 

number called the multiplier. 
Multiplication (mul-ti-pli-ka'shon). 29. 
Multiplier. See Multiplicand. 

Net proceeds (net pro'seds). The sum left from the sale of a note or 
other piece of property after every charge has been paid. 

Notation (no-ta/shon)/ L. notare, to mark. A system of written 
signs of things and relations used in place of common language. 

Note (not). L. nota, mark. A written or printed paper acknowledg- 
ing a debt and promising payment. 

Number (num'ber). L. numerus, a number. An abstract number is 
the ratio of one quantity to another of the same kind. See 
Abstract, Concrete. 

Numeration (nu-me-ra'shon). L. numerare, to count. The art of 
reading numbers. 



XIV HIGHER ARITHMETIC . 

Numerator (nu'me-ra-tor). The number, in a common fraction, 

which shows how many parts of a unit are taken. 
Ohm (om). Named after G. S. Ohm, a German electrician. The 

unit of electrical resistance. 101. 
Ordinate (or'di-nat). L. ordinare, to order. 68. 
Parallelepiped (par-a-lel-e-pip'ed or pl'ped). G. parallelos, parallel, 

+ epipedon, plane. A prism whose bases are parallelograms. 
Parallelogram (par-a-lel'o-gram). G. parallelos, parallel, + gramma, 

line. A quadrilateral whose opposite sides are parallel. 
Par value. L. par, equal. Face value. 
Per capita (per cap'i-ta). L., by the head. 
Per cent. L., by the hundred. 126. 
Percentage (per-sent'aj). (a) That portion of arithmetic which 

involves the taking of per cents, (b) The result from multiplying 

a number (called the base) by a certain rate per cent. 
Policy (pol'i-si). ML. politicum, a register. A contract of insurance. 
Poll tax (pol). ME. poll, head. A tax sometimes levied at so much 

per head of the adult male population. 
Premium (pre'mi-um). L. praemium, profit. (a) Amount paid to 

insurers as consideration for insurance, (b) Amount above par at 

which stocks, drafts, etc., are selling. 
Present worth of a sum. An amount which placed at interest at a 

given rate will amount to that sum in a given time. 
Prime number (prime). L. primus, first. An integer not divisible 

without a remainder by any integer except itself and unity. Two 

integers are prime to one another when they have no common 

divisor except unity. 
Prime meridian. See Longitude. 
Principal (prin'si-pal). L. princeps, first. A capital sum lent on 

interest. 
Prism (prizm). G. priein, to saw. A solid whose bases are parallel 

congruent polygons and whose sides are parallelograms. 
Problem (problem). G. problema, a question proposed for solution. 
Product (prod'ukt). L. pro-, forward, + ducere, to lead. The result 

from multiplying one number by another. 
Proper fraction. A common fraction whose terms are positive, and 

whose numerator is less than its denominator. 
Proportion (pro-por'shon). L. pro, before, + portio, share. 96. 
Pyramid (pirVmid). G. pyramis, a pyramid. A solid contained by 

a plane polygon as base, and other planes meeting in a point. 



DEFINITIONS AND ETYMOLOGIES. XV 

Pythagorean theorem (pi-thag'o-re-an). A theorem first proved by 
Pythagoras. 68. 

Quadrilateral (kwod-rl-lat'e-ral). L. quatuor, four, + latus, a side. 

A four-sided plane figure. 
Quantity (kwon'ti-ti). L. quantus, how much ? The being so much 

in measure or extent. 
Quotient (kwo'shent). L. quotiens, how many times ? The number 

which taken with the divisor as a factor produces the dividend. 

Radius (ra'di-us). L., rod. A line from the center of a circle to the 
circumference. 

Rate per cent. 127. 

Ratio (ra'shio). L., a reckoning. 87. 

Reciprocal numbers (re-sip'ro-kal). L. reciprocus, alternating. Two 
numbers which multiplied together make unity. 

Rectangle (rek'tang-gl). L. rectus, right, + angulus, angle. A quad- 
rilateral all of whose angles are right angles. 

Rectangular parallelepiped. A parallelepiped all of whose faces are 
rectangles. 

Reduction (re-duk'shon). L. re-, back, + ducere, to bring. Chang- 
ing the denomination of numbers. Reduction ascending, changing 
to a higher denomination, as from 144 inches to 12 feet. Reduc- 
tion descending, changing to a lower denomination. 

Registered bond (rej'is-terd). A bond bearing the name of the owner, 
the name and residence being registered on the books of the cor- 
poration issuing the bond. 

Remainder (re-man'der). L. re-, back, + racmere, to stay. The same 
as Difference. 

Root (rot or rut). ME. roote, root. The root of a number is such a 
number as, when multiplied into itself a certain number of times, 
will produce that number. For more extended definition, see Cube 
Root, Square Root. 

Series (se'rez or se'ri-ez). L. series, a row, from serere, to join 

together. 104. 
Share (shar). AS. scercm, to cut. One of the whole number of equal 

parts into which the capital stock of a corporation is divided. 
Significant figure (sig-nif i-kant). L. signum, a sign, + /acere, to make. 

The succession of figures in the ordinary notation of a number, 

neglecting all the ciphers between the decimal point and the figure 

not a cipher nearest to the decimal point. 



XVI HIGHER ARITHMETIC. 

Solid (sol'id). L. solidus, firm. Any limited portion of space. 

Specific gravity (spe-sif'ik grav'i-ti). L. species, kind, +/acere, to 
make. 92. 

Sphere (sfer). G. sphaera, a ball. A solid bounded by a surface 
whose every point is equidistant from a point within the solid, 
called the center of the sphere. 

Square (skwar). L. quatuor, four. (a) An equilateral rectangle, 
(b) The second power of a number, (c) To raise a number to the 
second power. 

Square root. 34. 

Standard time (stan'dard). ML. standardum, standard. A system 
of uniform time for a given section of country. 85. 

Stere (star). G. stereos, solid. A cubic meter; 35.31 cu. ft. 

Stock (stok). AS. stoc, post, trunk. The share capital of a corpora- 
tion. 

Subtraction (sub-trak'shon). L. sub, under, + trahere, draw. The 
operation of finding the difference between two numbers. See 
Difference. 

Subtrahend (sub'tra-hend). The number subtracted from the minuend. 

Sum (sum). L. summa, the highest part. See Addition. 

Surd (serd). L. surdus, deaf. A number not expressible as the ratio 
of two integers. 

Surface (ser'fas). L. superficies, the upper face. The bounding or 
limiting parts of a solid. 

Terms of a fraction (terms). L. terminus, limit. The numerator and 
denominator together. 

Theorem (the'o-rem). G. theorema, a sight. A statement of a truth 
to be demonstrated. 

Thermometer (ther-mom'e-ter). G. therme, heat, + metron, measure. 
An instrument by which temperature is measured. 

Trapezoid (tra-pe'zoid). G. trapeza, table, + eidos, form. A quad- 
rilateral having two parallel sides. 

Triangle (trrang-gl). L. tres, three, + angulus, angle. A three-sided 
plane figure. 

Unit (u'nit). L. unus, one. Any standard quantity by the represen- 
tation and subdivision of which any other quantity of the same 
kind is measured. 

Volt (volt). From Volta, an Italian physicist. The unit of electro- 
motive force. 101. 

Volume (vol'um). L. volvere, to roll round. Solid contents, 



SYMBOLS AND ABBREVIATIONS. 



THE following are used in this work, and are inserted 
here for reference. Other symbols are explained as needed. 
For historical note, see p. 43. 



E.g., 



I.e., 
ex., 
ax., 
th., 



Latin exempli gratia, for 

example. 

Latin id est, that is. 
exercise, 
axiom, 
theorem, 
g.c.d., greatest common divisor, 
l.c.m., least common multiple. 
Tt, Greek letter it, symbol for 

3.14159 

v, since. 

.-., therefore. 

%, per cent, hundredth, hun- 

dredths. See p. 126. 
+, plus, symbol of addition 
and of positive numbers. 
, minus, symbol of subtrac- 
tion and of negative 
numbers. 

plus or (and) minus, 
and absence of sign be- 
tween letters, denote 
multiplication, 
and fractional form de- 
note division, 
ratio, a special form of 

division, 
equal or equals, 
approaches as a limit. See 
p. 108. 



, 
x, ', 



: : , sometimes used in propor- 
tion as a sign of equality. 

>, is (or are) greater than. 

<C, is (or are) less than. 

:j, does (or do) not equal. 

;>, is (or are) not greater than. 

<, is (or are) not less than. 

, and so on. 

a- 2 , a- 1 , a, a 1 , a 2 , a n , indi- 
cate powers. See pp. 2, 
32. 

V~ and the exponent indicate 
square root. See p. 33. 

/- 1 

V and the exponent - indicate 
n 

nth root. 

( ), , symbols of aggregation, 
log, colog, antilog, see pp. 111- 

118. 

For abbreviations of common 
measures, see pp. 52, 53. For 
abbreviations of metric meas- 
ures, see pp. 60, 61. 

n' is read n-prime. 

fi " / sub one, or /-one. 

t n u t sub n, or (if there can 
be no misunderstand- 
ing) t-n. 



H1GHEE AEITHMETIO. 

CHAPTER I. 
Notation and the Fundamental Operations. 



I. WRITING AND READING NUMBERS. 

THE universal notation among civilized nations at the 
present time is the Hindu or Arabic, the symbols of which, 
except the zero, originated in India before the beginning 
of the Christian era, and seem to have been the initial 
letters of the early numerals. The system derives its in- 
trinsic importance, however, from the zero, which renders 
possible the distinctive feature known as place value. 
Thus, in the number 302, the 3 stands for hundreds 
because it is in hundreds' place, a fact which could not 
be conveniently indicated without the symbol or its 
equivalent. The zero ' appeared about the fifth century 
A.D., and somewhat later the Arabs, coming from the East 
after the conquest of Spain, brought the new system with 
them. About the year 1200 these Hindu numerals began 
to be known in Christian Europe, but it was not until the 
fifteenth century that they were generally taught and used. 
The decimal point appeared about the opening of the 
seventeenth century, and through its influence the sub- 
ject of arithmetic, both pure and applied, has materially 
changed. The extent of this change may be seen in the 
number of cases in which the decimal fraction is used 
to-day. 



2 HIGHER ARITHMETIC. 

The Roman numerals were in common use in Europe 
prior to the introduction of the Hindu system. As now 
written they have changed considerably from the form 
used at the beginning of the Christian era. In America 
they are at present employed chiefly in numbering the 
chapters of books, and hence are rarely used beyond one 
or two hundred. The old custom of printing the number 
of the year in Roman notation on the titlepages of books 
has practically ceased. 

In modern science numbers are often used which con- 
tain several zeros, for the reason that absolute accuracy of 
measurement is generally impossible. Thus, it is said 
that the distance from the earth to a certain star is 
21,000,000,000,000 miles, but the distance even to within 
a billion miles is quite unknown. Similarly, the length 
of a wave of sodium light is said to be 0.0005896 of a 
millimeter, but the seventh decimal place is doubtful and 
the subsequent ones are unknown. The naming of these 
numbers is a matter of little importance, and the writing 
of them in full is usually unnecessary. Scientists often 
resort to an index notation, in which an integer, sometimes 
followed by a decimal, is multiplied by a power of 10. 
Thus, 21,000,000,000,000 may be written 2.1 X 10 13 , or 
21 X 10 12 . And since 10" 1 means 0.1, and 10~ 2 means 
0.01, etc., therefore 0.00000274 may be written 2.74 X 10" 6 . 

Since the index notation is now so extensively used in 
science, and since the limit of necessary counting in finan- 
cial affairs is met in the billion or trillion, no elaborate 
system of naming numbers is practically used. Attention 
should be paid, however, to the proper reading of the num- 
bers in common use, types of which are given in the exam- 
ples on p. 3. Thus, 123.4567 should be read, one hundred 
twenty-three and four thousand five hundred sixty-seven 
ten-thousandths. 



NOTATION AND FUNDAMENTAL OPERATIONS. d 

Exercises. 1. What are the various names given to the symbol ? 

2. Read the numbers 0.0002, 0.00004, 0.400. 

3. Also, 0.123, 100.023. 

4. Also, 0.1246, 1200.0046. 

5. In the numbers XV and 15, why are the Hindu characters said 
to have a place value and the Roman not ? 

6. Express in the index notation the numbers in the following 
statements : 

(a) In a cubic centimeter of air there is 0.00001114 of a grain of 
water. 

(6) A centimeter is 0.0000062138 of a mile. 

(c) The distance to the sun is 93,000,000 miles. 

(d) The distance from the sun to Neptune is 2,788,800,000 miles. 

7. Express in the common notation the numbers in the following 
statements : 

(a) The distance from the equator to the pole is 39.377786 X 10 7 
inches. 

(6) The earth's polar radius is 6.35411 X 10 8 centimeters. 

8. A syndicate is to bid on some government bonds ; to how many 
decimal places should they express their bid per $100 if they bid for 
$10,000 worth? $1,000,000 worth? $25,000,000 worth ? $100,000,000 
worth ? 

II. CHECKS. 

A check on an operation is another operation whose 
result tends to verify the result of the first. E.g., if 
11 7 = 4, then 7 + 4 should equal 11 ; this second result, 
11, verifies the first result, 4. 

The verification is usually incomplete. If, as is said, 
"the result does not check," there must be an error in 
(1) the original operation, (2) the check, or (3) both. If, 
on the other hand, the result does check, there may have 
been an error in one operation which just balanced the 
other. Hence a check makes it more or less improbable 
that an error remains undiscovered. The secret of accu- 
rate computation largely lies in the knowledge and the 
continued use of proper checks. 



HIGHER ARITHMETIC. 



III. ADDITION. 

In adding the annexed column, the computer should say 
to himself, " Five, fourteen, eighteen ; one, seven, 
fourteen; nine, twelve" thus omitting all super- 374 

fluous words. Bookkeepers, whose business leads 69 

to rapid addition, omit much that would seem 805 

necessary to the student, and not infrequently 1248 
add two columns at once, a power gained only 
by practice in their profession. 

The most practical way to check addition is to perform 
the operation a second time. Experience shows, however, 
that if the operation is performed in the same way the 
mind tends to fall into the same error. Hence it is better 
to add the numbers in the reverse order. 

It is unnecessary to give many exercises in addition, 
since the student who finds himself in need of practice 
can easily prepare them. 

Exercises. 1. As an exercise, it is convenient to prepare a table 
of multiples of some number in this manner : Write any number, as 
4197, on paper, and the same number on a small card, 4197 | ; place 
the card above the number and add, thus giving 2 X 4197 ; slide the 
card down and add again, thus giving 3 X 4197, and so onto 10 X 4197, 
when the work checks if the result is 41,970. 

2. In the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, , each 

number after the first two is obtained by adding the two preceding 
numbers ; calculate the 30th number of the series. 

3. Add 15, 214, 3962, 9984, 9785, 6037. Check. 

4. Add 201, 76, 435, 7726, 8687, 8812, 8453, 1162. Check. 

5. What is the sum of the first two odd numbers, i.e., 1 + 3 ? of 
the first 3 ? of the first 4 ? of the first 10 ? of the first 20 ? From 
these results, what would be inferred as to the sum of the first 100 
odd numbers ? 

6. Explain why 234 + 859 is 1000 more than the difference between 
234 and 141. Also why 4396 + 8501 is 10,000 more than the difference 
between 4396 and 1499. 



NOTATION AND FUNDAMENTAL OPERATIONS. 5 

IV. SUBTRACTION. 

There are three common methods of subtraction. In 
the annexed example, we may say, 

(1) 5 from 14, 9 ; 2 from 2, ; 3 from 12, 9 ; 1234 

(2) 5 from 14, 9 ; 3 from 3, ; 3 from 12, 9 ; 325 

(3) 5 and 9, 14 ; 2 and 1 and 0, 3 j 3 and 9, 12. 909 
Each of these three methods is easily understood. The 

first is the simplest of explanation, and hence it is generally 
taught to children. The second is slightly more rapid than 
the first. But the third, familiar to all as the common 
method of " making change," is so much more rapid than 
either of the others that it is recommended to all computers. 
Since subtraction is the inverse of addition, the simplest 
check is the addition of the subtrahend and difference j the 
sum should equal the minuend. 

Exercises. 1. If not entirely familiar with the third method 
above given, use it with enough problems to become so, and state the 
reason of its advantage in rapidity over the others. 

2. In subtracting 34,256 from 100,000, show that the subtraction 
can easily be made from the left by taking each digit from 9, except 
the 6, which must be taken from 10. 

3. As in Ex. 2, show how to subtract 27,830 from 100,000 ; also 
948,900 from 1,000,000. 

In Exs. 2 and 3, the results are called the complements of the given 
numbers, because they complete the next higher power of 10. 

4. Show that the difference between 1234 and 5612 may be found by 
adding the complement of 1234 to 5612 and then subtracting 10,000. 

5. Does a b equal a + (10 6) 10 ? Does a b equal a + 
(10" 6) 10" ? Show that this proves that the method of subtrac- 
tion given in Ex. 4 is general. 

6. Is there any advantage in subtracting by means of adding the 
complement of the subtrahend in a case like 521 173 ? 

7. Solve the problem 6872 - 4396 + 342 - 896 - 243 + 750 by 
adding the proper complements and finally subtracting the proper 
powers of 10. Is there any advantage in using the method of com- 
plements in this case ? 



b HIGHER ARITHMETIC. 

V. MULTIPLICATION. 

The multiplication table is usually learned to 10 X 10. 
This is all that is necessary for practical purposes. It is 
often convenient, however, to perform quickly multiplica- 
tions with certain larger numbers. There are many rules 
for such operations, most of them of little practical value. 
Some are, however, quite useful, and these are given. 

To multiply by 5, 25, 33-J-, 125 is the same as to multiply 
by i^ ioo.^ 10 Q_ } iOg--- It is easier to multiply by J^ Q , that 
is, to add a cipher and divide by 2, than to multiply by 5, 
and similarly for the other cases. Also, to divide by 5 is 
the same as to divide by *- or to multiply by T 2 Q, and it is 
easier to multiply by 2 and divide by 10 than to divide by 
5. These short processes have especial value because 5%, 
12.5%, 25%, 33-J-%, 50% are common rates in business and 
scientific problems. 

To multiply by 9 is to multiply by (10 1). Hence, it 
suffices to annex a cipher and subtract the original number 
from the result, a method not always more convenient than 
ordinary multiplication. 

To multiply by 11 is to multiply by (10 + 1). Hence, it 
suffices to annex a cipher and add the original number ; or, 
what is more convenient, to add the digits as in the follow- 
ing example : 11 X 248, 

8 + = 8, 4+8 = 12, 2 + 4 + 1 = 7, 2 + = 2; 
therefore, the result is 2728. 

To multiply by numbers differing but little from 10". 
For example, to multiply by 997 is to multiply by (10 3 3) ; 
that is, to annex three ciphers and subtract 3 times the 
multiplicand. 

E.g., 995 X 1474 = 1,474,000 - i of 14,740 

= 1,474,000 - 7370 = 1,466,630. 



NOTATION AND FUNDAMENTAL OPERATIONS. 7 

When two factors lie between 10 and 20 the product 
is readily found as follows : 

14 Xl6 = 10(14 + 6) + 4 X 6 = 224. 
To prove that this process is general : 

(1) Any number from 10 to 19 may be represented by 

10 + a, 10 + &, 

(2) (10 + a) (10 + b) = 100 + 10 -a + 10 1 + ab, 

= 10 (10 + a + b) + ab. 

To square numbers ending in 5. While not as practical 
as the problems already given, this has some value. The 
method is illustrated by 65 2 ; it is merely necessary to say, 

6 X 7 = 42, 5 X 5 = 25, .-. 65* = 4225. 
To prove that this process is general : 

(1) Any number ending in 5 may be represented by 
10 a + 5, where a may equal 0, 1, 2, 9, 10, 

(2) (10 a + 5) 2 = 100 a 2 + 100 a + 25, 

= 100 a ( + !) + 25. 

(3) That is, the result ends in 25, and the number of 
hundreds is a (a + 1). 

Applications of the formulas for (a + 6) 2 , ( + &)( &). 
In a few problems there is an advantage in recalling the 
identities (a + fl) = a a + 2 aft + 2 , 

(a + l>)(a 1) = a 2 l\ 

For example, 62 2 = 3600 + 240 + 4 =3844. 

23 X 17 = (20 + 3) (20 3) = 391. 

Oral Exercises. 1. Multiply 12,564 by 5; by 25. 

2. Multiply 4239 by 0.33*. 

3. Multiply 5148 by 5; by 2.5 ; by 33*; by 12.5. 

4. Square 25, 35, 45, 95, 105, 115. 

5. Multiply 14 by 19; 17 by 15 ; 16 by 18; 12 by 19 ; 13 by 17. 

6. Multiply 22 by 99 ; by 98 ; by 97. 

7. Multiply 12,345 by 11. 

8. Explain the short process of dividing by 33* ; by 125. 



8 HIGHER ARITHMETIC. 

Arrangement of work. In multiplication, there is no 
practical advantage in beginning with, the 
lowest order of units of the multiplier ; 437.189 

in fact, there is a decided advantage in 26.93 

beginning with the highest order, as is 8743.78 

clearly apparent in approximations in 2623.134 

multiplication. The arrangement of work 393.4701 

would then be as shown in the annexed 13.11567 

example. Since 20 X 0.009 = 0.18, the 11773.49977 
position of the decimal point is at once 
known, and the rest of the process is apparent. 

Approximations in multiplication are frequently desired. 
This arises from the fact that perfect measurements are 
rarely possible in science, and that results beyond two or 
three decimal places are seldom desired in business. Thus, 
if the radius of a wheel is known only to 0.001 inch, it is 
not possible to compute the circumference 
with any greater degree of accuracy ; hence, 10.48 
labor would be wasted in seeking a product 3.1416 

to more than three decimal places. In 31.44 
such cases all unnecessary work should be 1.048 

omitted, as in the annexed example. This 0.419 . 

represents the multiplication in the solu- 0.010 . . 

tion of the following problem : To find the 0.006 . . . 

circumference of a steel shaft of which the 32.92 
diameter is found by measurement to be 
10.48 centimeters. Since it was practical to carry the 
original measurement only to 0.01 centimeter, the result 
need be sought only to 0.01. In order to be sure that the 
result is correct to 0.01, the partial products are carried to 
0.001, and in multiplying by 0.04, for example, the effect 
of the fourth decimal place on the third is kept in mind. 

Checks. The best check on multiplication is the " cast- 
ing out of nines," explained in Chap. II. 



NOTATION AND FUNDAMENTAL OPERATIONS. 9 

Exercises. 1. Multiply 42.3531 by 3.1416, carrying the result 
to 0.001, that is, " correct to 0.001." 

2. Multiply 126 by 0.3183, correct to 0.1. 

3. If V2 = 1.414 , find the values of 4.324 V, 0.057 V2, and 

8.346 V2, correct to 0.01. 

4. If V3 = 1.732 , find the values of 2.35 V, 42.89 V, and 

0.869V3, correct to 0.1. 

5. Find V2 VI, correct to 0.01. 

6. Find the interest on $1525.75 for one year at 6|%, correct to 
$0.01. 

7. If the circumference of a circle is 3.1416 times the diameter, 
and if the radius of the earth is found by measurement to be 
6,378,249.2 meters, find the circumference to the necessary number 
of decimal places, the earth being supposed spherical. Express the 
result in the index notation. 

8. Similarly, find the circumference of a wheel the diameter of 
which is found by measurement to be 6.3 ft. 

9. Similarly, find the circumference of a shaft the radius of which 
is found by measurement to be 4.32 in. 

10. If 1 yard is found by measurement to be 0.914 of a meter, 
find, to the necessary number of decimal places, the number of meters 
in 23.463 yds. 

11. Similarly, if 1 meter is found by measurement to be 3.28 feet, 
find the number of feet in 3.476 meters. 

12. If the number of cubic units of volume in a sphere is f of 3. 1416 
times the third power of the number of units of radius, find the volume 
of a sphere whose radius is found by measurement to be 3.27. 

13. Find the value of 9 X 12,345,678 + 9 ; of 9 X 1,234,567 + 8 ; 
of 9 X 123,456 + 7. 

14. Find the value of 9 X 98,765,432 + ; of 9 X 9,876,543 + 1 ; 
of 9 X 987,654 + 2. 

15. Find the value of 8 X 123,456,789 + 9 ; of 8 X 12,345,678 + 8 ; 
of 8 X 1,234,567 + 7. 

16. Show that 8,212,890,625 2 terminates in 8,212,890,625. 

17. If a railway uses a freight car belonging to another company 
it pays the owner 0.6 of a cent a mile for the distance run ; during one 
year the freight cars of the United States were used in this way over a 
total distance of about 12,000,000,000 miles. How much rental did 
they earn their owners ? 

18. Prove that to multiply by 625 one may move the decimal point 
four places to the right and then divide by 2 four times. 



10 HIGHER ARITHMETIC . 



VI. DIVISION. 

In division there is an advantage in placing the quotient 
above the dividend. The decimal point is then easily fixed, 
although it is unneces- 
sary to carry it through 
the work. 

In case a decimal 15040.0 

point appears in both I25fifi*4. 

dividend and divisor, it 247360 

is better to multiply 219912 

each by such a power 
of 10 as shall make the 

divisor integral. Thus, 

. ^ oo no 23.152 remainder. 

in the case of 32.92 -f- 

3.1416 it is better to multiply both numbers by 10 4 , and 
divide 329200 by 31416, as above. 

The work may be further abridged 10.478 

by omitting partial products and 31416)329200 
decimal points, keeping only the 15040 

partial dividends, as here shown. 24736 

It is advisable in extended cases 27448 

of division to prepare a table of 23152 

multiples of the divisor, as in the 
following division of 4,769,835 by 291 : 

1 291 

2 582 16391 

3 873 4769835 

4 1164 1859 

5 1455 1138 

6 1746 2653 

7 2037 345 

8 2328 54 remainder. 

9 2619 



NOTATION AND FUNDAMENTAL OPERATIONS. 11 

Approximations in division. In division as in multipli- 
cation, approximations are often necessary. For example, 
if the circumference of a shaft is found by measurement 
to be 32.92 centimeters and it is required to know the 
diameter, it would be a waste of time to attempt to find 
the diameter beyond 0.01. Since 10's divided by 10,000's 
< 0.01's, the last two figures of the dividend will not 
affect the quotient within two decimal places and hence 
may be neglected. Hence, also, the divisor may be con- 
sidered as 3142 and may be continually contracted. The 
process is apparent by first examining the complete form 
in the example below. The student should note how much 
better for practical purposes the last form is than the 
others, and he is recommended to become so familiar with 
it as to use it in all cases where only approximate results 
are required. 

10.48 
31416)329200 

3142 = approximately 10 X 3141(6) 



126 = 0.4 of 314(16) 

24 
24 = 0.08 of 31(416) 

10.48 

31416)329200 
150 
24 

Checks. The best check on division is the "casting 
out -of nines " explained in Chap. II. Since the dividend 
equals the product of the quotient and divisor, plus the 
remainder if any, the work may also be checked by one 
multiplication and one addition, 



12 HIGHER ARITHMETIC. 

Exercises. 1. Divide 42,856,731,275,834 by 574,238, correct to 
the tens' place. 

2. Divide 100 by 3.1416, correct to 0.01. 

3. Divide 5,080,240 by 40,467, correct to 0.1. 

4. Divide 1 by 3.14159, correct to 0.001. 

5. The population of British India is about 225,000,000, and the 
area is about 965,000 square miles ; what is the average population 
per square mile, to the nearest unit ? 

6. In a certain year it cost $357,231,799 to pay the expenses of the 
United States government, which was $5.346 to each person ; what 
was the population in that year, to the nearest 1000 ? 

7. In a certain year the revenue of the United States government 
was $403,080,983, which was $6.577 to each person; what was the 
population in that year, to the nearest 1000 ? 

8. A hectare is 2.471 acres, and a liter is 61.027 cubic inches ; 
express the rainfall of 1 liter per hectare in cubic inches per acre, 
correct to 0.1. 

9. Knowing that the circumference of a circle is 2 X 3.14159 X the 
radius, find, correct to 0.1, how many revolutions a mill-wheel 12 feet 
in diameter makes per minute when the speed of the periphery is 
6 feet per second. 

10. How many revolutions per mile are made by a locomotive 
drive-wheel 4.5 feet in diameter ? (Correct to units.) 

11. 84.25 liters of water are drawn through a pipe every 4.5 
minutes from a tank containing 23,711 liters ; how long will it take 
to empty the tank ? (Correct to 1 minute.) 

12. If 41 liters of water weigh as much as 51 liters of alcohol, and 
1 liter of water weighs 1 kilogram, how much will 1 liter of alcohol 
weigh ? (Correct to 0.01 kilogram.) 

13. The horse-power of an engine is usually calculated by the 

formula , where p, Z, a, n are abstract numbers representing 
oo,000 

the pressure in pounds per square inch on the piston, the length of 
the stroke in feet, the area of the piston in square inches, and the 
number of strokes per minute. Calculate the horse-power, to the 
nearest unit, of each of these engines : 



(a) p = 20, 


1= 6, 


a= 400, 


n = 60. 


(6) p = 8, 


J = 11, 


a = 3600, 


n = 40. 


(c) p = 25, 


1= 3, 


a= 100, 


n = 90. 


(d) p = 18, 


1= 5, 


a= 200, 


n = 50. 



NOTATION AND FUNDAMENTAL OPERATIONS. 13 

Axioms. There are a number of general statements 
of mathematics the truth of which may be taken for 
granted. Such statements are called axioms. 

The following are the axioms most frequently used in 
elementary arithmetic and algebra, the first, second, third, 
sixth, and seventh being especially important : 

1. Numbers which are equal to the same number or to 
equal numbers are equal to each other. 

If 5 x = 3, and 1 + x = 3, then 5 x = 1 + x. 

2. If equals are added to equals, the sums are equal. 

If x 2 = 7, then x 2 + 2 = 7+2, or x = 9. 

3. If equals are subtracted from equals, the remainders 
are equal. 

If x + 2 - 9, then x - 7. 

4. If equals are added to unequals, the sums are unequal 
in the same sense. 

If x + 2 > 8, then x + 2+5>8 + 5. 

5. If equals are subtracted from unequals, the remainders 
are unequal in the same sense. 

If x + 5<16, then x< 11. 

6. If equals are multiplied by equal numbers, the products 
are equal. 

If ~ = 6, then x = 18. 
o 

7. If equals are divided by equals, the quotients are 
equal. 

If 2 x = 6, then x = 6 + 2 = 3. 

8. Like powers of equal numbers are equal. 
If x = 5, then x 2 = 25. 

9. Like roots of equal numbers are arithmetically equal. 

If x 2 = 25, then x = 5. (From algebra it should be remembered 
thatx= 5.) 



14 HIGHER ARITHMETIC. 

Fundamental laws of elementary operations. There 
are also certain fundamental laws of number, the strict 
proof of which for cases involving fractions, surds, nega- 
tive numbers, etc., is properly a part of algebraic analysis. 
Since their treatment is too advanced for an elementary 
work, their validity is here assumed. They are not, how- 
ever, axioms, because they are not generally taken for 
granted in mathematics. 

These laws, as well as the axioms on p. 13, are so fre- 
quently used in subsequent discussions that their formal 
statement is necessary. They are as follows : 

1. The associative law for addition and subtraction, that 
the terms of an expression may be grouped in any way 
desired. 

E.g., a + b-c + d = a + (b-c) + d = (a + b)-(c-d) = 

2. The commutative law for addition and subtraction, that 
these operations may be performed in any order desired. 

E.g., a + b c + d = a c + d + b = d c + a + b= 

3. The associative law for multiplication and division, 
that these operations may be grouped in any way desired. 

E.g., a-b-c + d-e = a-(b'c) + (d-7- e) = a-b-(c-r d)-e = 

4. The commutative law for multiplication and division, 
that these operations may be performed in any order 
desired. 

E.g., a-b-r-c = a + c-b = b + c-a= 

Of course it is absurd to say that 72 times $3 is the same as 
$3 times 72, since the latter has no meaning by the common idea of 
multiplication. All that the commutative law asserts is that 72 $3 
= 72 3 $1 = 3 72 $1 = 3 $72. 

5. The distributive law for multiplication and division, 

that m(a b) = ma mb, and that - = 

m mm 



CHAPTER II. 
Factors and Multiples. 

I. TESTS OF DIVISIBILITY. 

IN speaking of factors and multiples, only integers are 
considered. Thus, 2 and 3.5 are not considered factors of 
7, nor is 9 considered a multiple of 2, although 4J X 2 = 9. 

In practical computations, cancellation enters exten- 
sively, requiring a knowledge of the factors of numbers. 
There is no general process for determining large factors, 
and hence elaborate factor tables have been prepared for 
those who have extensive computations to make. But 
there are simple methods for determining small factors, 
methods not only valuable in a practical way, but also for 
the logic involved in their consideration. 

Fundamental theorems. The theory of factors and 
multiples depends largely on two theorems : 

I. A factor of a number is a factor of any of its multiples. 
1. Let n be any number of which/ is a factor, q being the quotient 



2. Then, since = 7, 

3. .-. - + - + ..... m times = q + q + ..... m times, Ax. 2 

4. Or -r mq. That is, / is a factor of mn, a multiple of n. 



16 HIGHER ARITHMETIC. 

II. A factor of each of two numbers is a factor of the sum 
or the difference of any two of their multiples. 



2. Then = aq, and = bq'. Th. 1 

3. Suppose an > bn', 

4. Then an ^ W = aq bq'. Axs. 2, 3 

That is, / is a factor of an 4- &M' and of an bn', the sum and 
the difference of two multiples of n and ri. 

Tests of divisibility. 

I. 2 is a factor of a number if it is a factor of the number 
represented by its last digit, and not otherwise. 

1. Any number has the form 10 a + 6, where 6 has any value from 
to 9 inclusive, and a has any value from and including 0. 

(E.g., 7036 = 10 X 703 + 6, and 3 = 10 X + 3.) 

2. 2 is a factor of 10, and hence of 10 a. Th. 1 

3. .-. 2 is a factor of 10 a + b if it is a factor of 6. Th. 2 

4. Otherwise 2 is not a factor of 10 a + 6, for Wa + b - 5 a + |, 

7 

and - is not an integer. 

II. 4 i s a factor of a number if it is a factor of the num- 
ber represented by its last two digits, and not otherwise. 

Any number has the form 100 a + 10 6 + c, where, etc. The proof, 
which is similar to that of I, is left for the student. 

III. 8 is a factor of a number if it is a factor of the 
number represented by its last three digits, and not other- 
wise. 

The proof, which is similar to the proofs of I and II, is left for the 
student. 

IV. 5 is a factor of a number if it is a factor of the num- 
ber represented by its last digit, and not otherwise. 

The proof, which is similar to that of I, is left for the student. 



TESTS OF DIVISIBILITY. 17 

V. 9 is a factor of a number if it is a factor of the sum of 
the numbers represented by its digits, and not otherwise. 

1. Any number may be represented by 

a + 106+ 100 c + 1000 d + 10,000 e + 

where a represents the units' digit, 6 the tens', 

(E.g., in 7024, a = 4, b = 2, c = 0, d 7, e = 0, ) 

2. Or by a +96 + 6 + 99c + c + 999<Z + d + 9999e + e + 

3. Orby96 + 99c + 999d+ + a + b + c + d+ 

4. Or by a multiple of 9, plus the sum of the numbers represented 
by the digits. 

5. .-. 9 is a factor if it is a factor of the latter, and not otherwise. 

VI. 3 is a factor of a number if it is a factor of the sum 
of the numbers represented by its digits, and not otherwise. 

The proof, the first three steps of which are the same as the first 
three steps of V, is left for the student. 

VII. 6 is a factor of a number if 2 is a factor of the 
number represented by the last digit, and if 3 is a factor of 
the sum of the numbers represented by its digits, and not 
otherwise. 

For, since 6 = 2 X 3, if the number is divisible by 2 and 3, it is 
divisible by 6. 

VIII. 11 is a factor of a number if it is a factor of the 
difference between the sums of the numbers represented by the 
odd and the even orders of digits, and not otherwise. 

E.g., 14,619 is divisible by 11, since (1 + 6 + 9) - (4 + 1) is divis- 
ible by 11. 

Proof. 1. Any number may be represented by 

a-f 106 + 100 c + 1000 d-f- 10,000 e+ 

2. Or by a + 116-6 + 99c + c + 1001 d - d + 9999 e + e + 

3. Orby 116 + 99c + 1001 d + 9999 e + 

+ (a + c + e+ ) (b + d+ ) 

4. Or by a multiple of 11, plus the difference between the sums of 
the numbers represented by the odd and the even orders of digits. 

5. .-. 11 is a factor if it is a factor of the latter, and not otherwise. 

IX. There is no simple method of testing divisibility by 7. 



18 HIGHER ARITHMETIC. 

Exercises. 1. State a short test for divisibility by 12 and prove 
that it is correct. 

2. Similarly for 15. 

3. Similarly for 16. 

4. Similarly for 18. 

5. Prove that 4 is a factor of a number if it is a factor of the sum 
of the units' digit and twice the tens', or of the difference between 
them. 

6. Prove that 8 is a factor of a number if it is a factor of the sum 
of the units' digit, and twice the tens', and four times the hundreds'. 

7. Is 823 a prime number ? Why is it unnecessary to try factors 
above 23 hi answering this question ? What are the prime factors of 
13,168 ? 

8. Prove that every even number is of the form 2n, every odd 
number of the form 2n 1, and every number not divisible by 3 of 
the form 3 n 1. 

9. Prove that every prime number above 3 is of the form Gn 1. 

10. Prove that one of any two consecutive even numbers is divisible 
by 4. Hence, show that 24 is a factor of the product of any three 
consecutive numbers if the middle one is odd. 

11. Prove that the product of any three consecutive numbers is 
divisible by 6, and that the product of any five consecutive numbers 
is divisible by 120. 

12. By squaring 10 a + 6, and then letting b have the various 

values 0, 1, 2, 9, prove that every square number is of the form 

5n or 5n 1. Hence, show that an integer cannot be a square 
number unless, when divided by 5, there is a remainder of 0, 1, 
or 4. 

13. What numbers below 19 (excluding 7, 13, 17) are factors of 
472,396,890,163 ? 

14. Similarly for 729,876,312 (excluding 7, 13, 14, 17). 

15. Prove that because 7 and 13 are factors of 1001 they are factors 
of 2002, 3003, 99,099, 100,100, 101,101 

16. Reduce to lowest terms, by cancellation, the fractions T ff f 7 , 
ff f , and ft**. 

17. Reduce to lowest terms, by cancellation, the fractions 

and rfftttflk- 

18. Reduce to lowest terms, by cancellation, the fractions 
tVW, and Hfttf 

19. Reduce to lowest terms, by cancellation, the fractions 
and T fiff 5 . 



CASTING OUT NINES. 19 



II. CASTING OUT NINES. 

The practical method of determining whether or not a 
number is divisible by 9 is as follows : Add the numbers 
represented by the digits and reject ("cast out") each 9 
as it is reached; the resulting number represents the 
remainder. 

Thus, to determine whether 124,763 is divisible by 9, say "3, 9 
(reject it), 7, 11 (reject 9), 2, 4, 5" ; therefore, the remainder is 5. 
Since the order of adding is immaterial, the eye usually groups the 9's 
at once and the remainder is easily detected. 

Check on addition by casting out nines. Since numbers 
are always multiples of 9 plus some 

remainder, they are of the type 9m-\-r. 9m -\-r 

By adding numbers of this type, the 9 ra f + r' 

sum is a multiple of 9, plus the sum 9 m" -f r" 

of the excesses. Therefore, the excess ' ' 

of nines in a sum is equal to the excess 9 (m + m' + ) 

in the sum of the excesses. r *' ~r r T* ' 

This check is too long to be of any 

value in addition alone ; it is, however, a necessary part 
of the check on division. 

Check on multiplication by casting out nines. Any two 

numbers may be represented by Qm-\-r 9 9 m f + r'. Their 
product is then represented by 9 2 mm' + 9 (mr 1 + m'r) -f rr', 
that is, by a multiple of 9, plus rr'. (Prove this by multi- 
plying 9 m + r by 9 m' + r 1 .) Since the excess of nines in 
this product is the excess in rr', therefore the excess of 
nines in any product equals the excess in the product of the 
excesses. 

E.g., 38 = 4x9 + 2 3 is the excess in 2 X 6, that is, 

51 = 5X9 + 6 the excess in the product of the two 

1938 = 215 x 9 + 3 excesses, 2 and 6. 



20 HIGHEK ARITHMETIC. 

Check on division by casting out nines. Since the 
dividend equals the product of the quotient and divisor, 
plus the remainder, the excess of nines in the dividend 
equals the excess in the sum of the excess in the product 
of the excesses of divisor and quotient, plus the excess in 
the remainder. 

E.g., 561,310,123 -=- 7654 = 73,335, with a remainder of 4033. 

.-. 561,310,123 = 7654 X 73,335 + 4033. 

.-. the excess in 561,310,123, which is 4, equals the excess in 4 X 3, 
which is 3, plus the excess in 4033, which is 1. 

Of course these checks fail to discover any error not 
affecting the excess of nines, as an interchange of digits, 
the addition of 9, etc., but such errors are rare. 

Exercises. 1. What is the remainder after dividing 4,236,987 
by 9 ? also 147,362 ? also 140,076,923 ? 

2. If 9,342,813 and 6,123,345 were added, would the sum be 
divisible by 9 ? 

3. Is 4,238,964,108 divisible by 9 ? by 3 ? by 6 ? by 11 ? 

4. If 12,345,601 were multiplied by 47,623,092 would the product 
be divisible by 3 ? by 9 ? by 11 ? 

5. Determine mentally which of the following products are incorrect 
by the test of casting out nines : (1) 41,376 X 87,147 = 3,605,794,272 ; 

(2) 11.1 X 307.05 = 3418.255; (3) 30,303 X 300,065 = 9,192,869,695; 
(4) 1213 = 1,771,561 ; (5) 444 X 31 = 13,764. 

6. Show that multiplication may also be checked by casting out 
threes. Apply this method to the products in Ex. 5. 

7. Similarly by casting out elevens. 

8. Is the product of the numbers 4398, 14,765, 900,427, and 2002 
divisible by 2 ? by 3 ? by 4 ? by 5 ? by 6 ? by 8 ? by 9 ? by 11 ? 

9. Determine mentally which of the following quotients are incor- 
rect by the test of casting out nines : (1) 865,432 -f- 2171 = 398 with 
1374 remainder; (2) 866,555-^5843 = 148 with 1891 remainder; 

(3) 4000 -r 23 = 173 with 21 remainder ; (4) 9012 -=- 173 = 52 with 
16 remainder. 

10. Show that division may also be checked by casting out threes. 
Apply this method to the quotients in Ex. 9. 



GREATEST COMMON DIVISOR. 21 



III. GREATEST COMMON DIVISOR. 

The subject of greatest common divisor has lost much of 
its practical value since the decimal fraction came into quite 
general use, during the eighteenth century. Formerly it 
was necessary to reduce a fraction like ^f f f to its lowest 
terms before it could be conveniently used in operations, 
e.g., added to another fraction. For this purpose, the 
greatest common divisor (here 59) was found and can- 
celled from each term. And since the greatest common 
divisor of such numbers is not easily found by inspection, 
the long division process (called from Euclid, who used it 
about 300 B.C., the "Euclidean method 7 ') was employed. 
This Euclidean method is now rarely met in practice, but 
as a piece of logical reasoning it is so valuable as to deserve 
a place in the review of arithmetic. 

The method of factoring may be illustrated by the fol- 
lowing example : Required the greatest common divisor of 
9801, 33,759, and 121,968. 

1. 9801 = 3 4 X11 2 . 

2. 33,759 = 3 >2 X II 2 X 31. 

3. 121,968 = 3 2 X II 2 X 2 4 X 7. 

4. And since the greatest common divisor is the greatest 
factor common to the three numbers, it is 3 2 X II 2 , or 1089. 

Exercises. 1. Find, by factoring, the g.c.d. of 153, 891, and 1008. 

2. Also of 32,760, 1170, and 1573. 

3. Also of 720, 336, and 1736. 

4. Also of 837, 1134, and 1377. 

5. Also of 187, 253, and 341. 

6. Also of 1331 and 4,723,598. 

7. Also of 231, 165, 451, 4004, and 2827. 

8. Also of 117, 143, 221, 338, and 650. 

9. Determine mentally that 49, 81, 121, and 4936 are prime to one 
another. 

10. Similarly for 429, 490, and 12,347. 



22 



HIGHER ARITHMETIC. 



4356)9801 

8712 4 

1089)4356 

4356 



The Euclidean or long division method may also be illus- 
trated by a single example, the one already considered. 

1. v the g.c.d. is contained 

in each number it ^j> 9801. 3 

2. The g.c.d. ^ 9801, v that 9801)33759 

is not a factor of 33,759. 29403 2 

3. v the g.c.d. is a factor of 
9801 and 33,759, it is a factor 
of 4356. P. 16, th. 2 

4. .'. the g.c.d. y> 4356 and 

it is 4356 if that is a factor of 

9801, 33,759, and 121,968. 

5. But the g.c.d. ^ 4356 112 
V that is not a factor of 9801. 1089)121968 

6. V the g.c.d. is a factor of 1089 
4356 and 9801, it is a factor 1306 
of 1089. P. 16, th. 2 1089 

7. .-. the g.c.d. y> 1089 2178 
and it is 1089 if that is a 2178 
factor of 4356, 9801, 33,759, 

and 121,968. 

8. v 1089 is a factor of 4356, it remains to find whether 
it is a factor of 9801, 33,759, and 121,968. 

9. v 1089 is a factor of itself and of 4356, it is a factor 
of 9801. P. 16, th. 2 

10. .-. it is a factor of 33,759. P. 16, th. 2 

11. And 1089 is a factor of 121,968, by trial. 

12. .'. 1089 is the greatest common divisor of the three 
numbers. 

Exercises. 1. Find, by the Euclidean method, the g.c.d. of 6961 
and 9976. 

2. Also of 8673 and 23,989. 

3. Also of 2827, 3341, and 11,565. 

4. Also of 5187, 14,421, and 3249. 



GREATEST COMMON DIVISOR. 23 

Abbreviations of the Euclidean form will readily suggest 
themselves. Only two will, however, be considered. 

1. If a factor is common to two numbers it must be a 
factor of their greatest common divisor. Hence, if seen, 
it may be suppressed in order to shorten the work, and 
introduced in the result. For example, the factor 2 in the 
problem below. 

2. If a factor of either number is not common to the 
other it cannot be a factor of their greatest common divisor. 
Hence, if seen, it may be suppressed in order to shorten 
the work. For example, the factor 3 2 in the problem 
below. 

Required the g.c.d. of 33,282 and 73,874. 

2133282 19 

1849 > 36937 

1844T l 

1806)1849 42 




Since a composite factor, like 9, may not be common to two 
numbers and yet may contain a factor, as 3, which is common, 
only prime factors should be rejected. 

Exercises. 1. Find, by the Euclidean method, suppressing factors 
whenever it is advantageous, the g.c.d. of 845,315 and 265,200. 

2. Also of 4,010,401 and 4,011,203. 

3. Also of 16,897 and 58,264. 

4. Also of 40,033 and 129,645. 

5. Also of 29,766 and 208,362. 

6. Also of 376, 940, 1034, and 1081. 

7. Reduce to lowest terms the fractions TT and 

8. Also the fractions \%\\ and ff \\. 

9. Also the fractions fff f and y^V 
10. Also the fractions T 9 <jWy and ffff . 



24 HIGHER ARITHMETIC. 

IV. LEAST COMMON MULTIPLE. 

In adding common fractions it is necessary to reduce 
them to fractions having a common denominator, and pref- 
erably to fractions having their least common denominator. 
Hence, when common fractions were largely used, this sub- 
ject was of great importance. The extensive use of the 
decimal fraction at the present time has, however, rendered 
unnecessary any such elaborate treatment of the subject as 
the earlier works present. As in the case of the greatest 
common divisor, the interest is now in the theory rather 
than in the practical applications. 

The method of factoring may be illustrated by the fol- 
lowing example : Eequired the least common multiple of 
9801, 33,759, and 121,968. 

1. 9801 = 3 4 X II 2 . 

2. 33,759 = 3 2 X II 2 X 31. 

3. 121,968 = 3 2 X II 2 X 2 4 X 7. 

4. And since the least common multiple contains all 
three numbers, and no unnecessary factors, it contains the 
factors 3 4 , II 2 , 2 4 , 7, and 31 and no others, and therefore is 
34,029,072. 

The greatest common divisor may be used in finding the 
least common multiple. Thus, in the above example : 

1. The g.c.d. of 9801, 33,759, and 121,968 is 1089. 

2. .'. this factor enters once and no more into the l.c.m. 

3. .'. this may be suppressed from any two of the 
numbers, as from 9801 and 121,968, leaving 9 and 112, 
and the other number may be multiplied by these two 
factors. 

4. And since 1089 contains all the common factors, 
9 X 112 X 33,759, or 34,029,072, contains all the factors 
of the three numbers, without repetition, and is therefore 
their least common multiple. 



LEAST COMMON MULTIPLE. 25 

Exercises. 1. What is meant by one number being a divisor of 
another ? a common divisor of two or more numbers ? the greatest 
common divisor of two or more numbers ? 

2. What is meant by one number being a multiple of another ? 
a common multiple of two or more numbers? the least common 
multiple of two or more numbers ? 

3. State the relative advantages of the two methods given for find- 
ing the greatest common divisor of several numbers. 

4. Similarly for finding the least common multiple of several 
numbers. 

5. Explain what is meant by a prime number ; by two numbers 
being prime to each other. Is 2. 5 a prime number? Are 8 and 21 
prime numbers ? prime to each other ? 

6. Find the least common multiple of 100, 101, and 103. 

7. Also of 100, 240, and 515. 

8. Also of 376, 1034, and 1081. 

9. Also of 173,376 and 171,072. 

10. The l.c.m. of two numbers is 96 and one of the numbers is 6, 
what values may the other number have ? 

11. Find the g.c.d. and also the l.c.m. of 763 and 2071. How does 
the product of the g.c.d. and the l.c.m. compare with the product of 
the two numbers ? 

12. Similarly for 2033 and 8239. 

13. Similarly for 8321 and 9577. 

14. Prove that the product of the g.c.d. and the l.c.m. of any two 
numbers always equals the product of the numbers. 

15. Find two pairs of numbers whose g.c.d. is 12 and whose sum is 96. 

16. Of numbers below 100, what ones have with 360 the g.c.d. 4 ? 

17. Find the g.c.d. and the l.c.m. of 144, 176, and 272 ; also of 161, 
253, and 299. 

18. One of two cog wheels which work together has 21 cogs, and 
the other 11 ; if a certain cog of one wheel rests on a certain cog 
of the other, after how many revolutions of the smaller will they be 
in the same relative position? after how many revolutions of the 
larger ? 

19. Three steamers arrive at a certain port, the first every Mon- 
day, the second every 10 days, the third every 12 days ; they all 
arrive on Monday, May 1. When will (a) the first and second next 
arrive together ? (6) the first and third ? (c) the second and third ? 
(d) all three ? 



CHAPTER III. 
Common Fractions. 



THE decimal point began to be used about the beginning 
of the seventeenth century, but it was over a hundred 
years before the new decimal fractions were extensively 
taught. During the transition period the old style frac- 
tions were so generally used that they were distinguished 
from the others by the term " vulgar " or " common " 
fractions, a name which still remains, although the dec- 
imal form is now more generally employed. The word 
vulgar then meant common and is still used in England 
in speaking of these fractions, although not generally so 
employed in America. It should, however, be remem- 
bered that the decimal fraction is only a special kind of 
a common fraction written in a special way. Thus, 0.25 
is merely -flfy, or J; that is, it always has a denominator 
10", and this denominator is not expressed, but is indi- 
cated by the position of the decimal point. 

The student is already familiar with the various opera- 
tions involving fractions. He needs, however, to review 
the reasons included in these operations, reasons imper- 
fectly presented (if at all) in the primary school, or since 
forgotten. 

A fraction is one or more of the equal parts of a unit. 

Thus, the fraction f is two of the three equal parts of one, the frac- 
tion 0.5 is five of the ten equal parts of one, and the fraction -^ is 
seventeen of the three equal parts of one. 



COMMON FRACTIONS. 27 



2 52 
This definition is incomplete. It excludes such fractions as -r-r 

* 1 3 " 

f , , , 0.131313 , 3.14159 , etc. But it includes decimal 

a, 
fractions, and common fractions of the form , a and 6 being positive 

integers, and these are the ones practically used. More scientifically 
defined, a fraction is an expressed division ; but the treatment of the 
subject under this definition is so abstract as to be better adapted to 
more advanced works. 

The terms numerator (Latin, numberer, because it num- 
bers the parts) and denominator (Latin, namer, because it 
names the parts) need no definition. 

A fraction is called a proper fraction when the numer- 
ator is less than the denominator j otherwise an improper 
fraction. 

Fundamental properties of fractions. 

I. An integer may be expressed as a fraction with any 
given denominator. 

Thus, to express a as a fraction with the denominator b. 

1. '-'I =|i that is, ftftths. 

2. .-. a a X 6 6ths, or ab ftths, 

= a6 < 
~ b ' 

Hence, any integer may be considered a fraction with the denomi- 
nator 1, thus broadening somewhat the original idea of a fraction. 

II. An improper fraction may be reduced to an integer, 
or an integer plus a proper fraction. 

Thus, to express - 3 7 7 - as an integer plus a proper fraction. 

1. As $37 contains $7 5 times, with a remainder of $2, 

2. so - 3 y- contains } 5 times, with a remainder of ^. 

3. That is, in 3 r 7 - there are 5 units + f . 

The proof, though applied to a particular fraction, is evidently 
general. The same result could have been obtained by dividing the 
numerator by the denominator. 



28 HIGHER ARITHMETIC. 

III. Multiplying or dividing the numerator of a fraction 
by any number multiples or divides, respectively, the value 
of the fraction by that number. 

a x 6 6 

1. - = a X - , because there are a times as many cths as before, 
c c 



2. = - -j- a " " " -th " " u " " " 

c c a 

IV. Multiplying or dividing the denominator of a fraction 
by any number divides or multiplies, respectively, the value 
of the fraction by that number. 

1. - = -th of -i because if the unit is divided into ft times as 

CL X C ft C 

many parts, each part (and hence the fraction) is only -th as large. 

2. ; = a x - > because if the unit is divided into -th as many 
c ft c a 

parts, each part (and hence the fraction) is a times as large. 

V. The value of a fraction is not altered by multiplying 
or dividing both terms by the same number. 

Since n x r = \- > by III, and n X ^ = ^ by IV, 
bo rib o 

a na a na , 

n x T = n x > or -r = by ax. 7. 

b nb b nb J 



Addition and subtraction of fractions. 

I. When they have a common denominator, as -;-;> 
7 _ ad 

where 



As $a $6 = | (a 6), the unit $1 being the same, 

so - - = - the unit - being the same. 

add d 

II. When they have not a common denominator, they 
may be reduced to fractions having a common denominator 
(by fundamental property V), and preferably to fractions 
having the least common denominator. 



COMMON FRACTIONS. 29 

Multiplication of fractions. To multiply one number 
by another is to perform that operation upon the first 
which being performed on unity produces the second. 

Since the primitive notion of multiplication is the taking of a num- 
ber a certain number of times, as 5 times $2, and since it is meaning- 
less to take a number 3 inches times, multiplication is considered as 
an operation in which the multiplicand may be either abstract or 
concrete, but in which the multiplier is always abstract. 

To illustrate the definition, 2 X $3 = $6 ; since 1 is added to itself 
to produce the multiplier, $3 is added to itself to produce $6. 

Similarly in the case of =- X - To produce the multiplier = from 
1, 1 must be divided into 6 parts, and a of those parts taken. So to 
produce the product from the multiplicand - the fraction J must be 

divided into 6 parts (each being by IV), and a of those parts 

ac 
taken, giving 

The symbols of multiplication, X and , are usually read "of" 
after a pure fraction, but " times " after an integer or a mixed number. 
Thus, I- x $5 is read "f of $5," but 1-j- X $5 is read " 1| times $5," 
this being the abridged form for " once $5 and of $5." (See p. 42.) 



Division of fractions. Division may be defined as the 
operation of finding one of two factors, the quotient, when 
the product and the other factor are given. 

For example, 2 X $5 = $10, .-. $10 -=-$5 = 2, and $10 -f 2 = $5. 

To divide r by is, therefore, to find a quotient q, such that 



.-. = q, by multiplying equals by d, and dividing by c, by axs. 6 

oc 

and 7, and fundamental properties III and IV. 

It therefore appears that the quotient equals the product of the 
dividend and the reciprocal of the divisor, and can, therefore, be 
obtained by multiplication. Hence the familiar rule, "Invert the 
divisor and multiply," a rule that is always valid for abstract divisors. 



30 HIGHER ARITHMETIC. 

a 

Complex fractions of the form - c may be considered as the 
equivalent of - -r- -, and treated accordingly. 



And since nXX- nXX^- = nXX~-^- 
d c_ d b d b d d 

d 

nX l 
and n X - X = n X - , by def . of division ; 

nx 
a 

a a 



c c 

- n x - 
d d 

That is, fundamental property V applies to complex fractions as 
well as to simple fractions. 

Practical suggestions as to the treatment of fractions. 

I. Make free use of fundamental property V, multiplying 
or dividing the terms by the same number. 

For example, the fraction T \ 9 g 3 f should have the factors 9 and 11 
suppressed at once, the fraction reducing to T 7 ^. In the case of the 

complex fraction -^ the factor 8 should be introduced, the fraction 
reducing to f , or T 5 , an operation much simpler than division. 

II. In reducing to lower terms, it is best to reject simple 
factors at once, without attempting to find the greatest com- 
mon divisor by the long process. 

For example, in the case of the fraction f/-/ 7 above considered. 



III. Feel free to use the common fraction or the decimal 
as may be the more convenient in the computation in hand. 

For example, it is better to multiply or divide by \ than by 0.25, 
and by j- than by 0.125. But 0.2 is an easier operator than , and 
0.04 is easier than . 



COMMON FRACTIONS. 31 

Exercises. 1. Explain the reduction of 5f to - 3 T 7 . 

2. Also the reduction of Q to 7|. 

3. Perform at sight the following multiplications: 0.125 of 640, 
0.33i of 903, 0.25 of 500, and 0.5 of 720. 

4. Also the following divisions: 840^0.125, 69 -r 0.33, 200 
-r 0.25, and 68 -r 0.5. 

5. Reduce to lowest terms the fractions f f , /J^, $|j[$, and f f . 

6. Simplify^, |, y, and|- 

7. Add 2/31, ^f y ! 3\4> If ft- 3 ( Th e Actors of 4199 are 13, 17, 
and one other.) 

8. Also the fractions ifoi ? 259^ an( j ^59^ 

9. A + A of T 4 T + T 5 2 of T 4 T of A + & of A of A of f + T 5 of A 
of T 3 o of f of 8 = ? 

10. | of ^of^of ^=? 

11. 1|- x 2^ x 3 x 4^ x 5 = ? 

12. Divide f of | of 3^ by f of | of 4. 

13. Divide ^ of & by ^5 of ^. 

14. Of the three fractions |, ^f , and f f , which is greatest ? which 
least? 

15. Which is greater, f or f f ? | of | or f of f ? 

16. What is the effect of adding the same number to both terms 
of a proper fraction on the value of the fraction ? Prove it for the 

general case of T' where a < 6. 

17. Investigate the same for the fraction -rt when a > 6. 

18. Show that the fraction lies between the greatest and 

o ~r o "r / 

the least of the fractions f , f , f . 

19. A " magic square " is a square array of numbers such 438 
that the sums of the numbers in the rows, columns, and two 951 
diagonals are equal, as in the annexed illustration. Insert 270 
the fractions to complete the following magic square : 



If 

* 

H 



CHAPTER IV. 
Powers and Roots. 



THE cases are few in practical business where either 
involution or evolution is used. In scientific work, num- 
bers often have to be raised to powers, and roots have to 
be extracted, but the operations are usually performed 
with the help of tables of powers, roots, or logarithms. 
The value of the subject may, therefore, be said to lie 
largely in the exercise of the reasoning powers. Hence, 
in the present chapter more attention is directed to the 
reasons for the various steps than to short methods of 
securing results. 

I. INVOLUTION. 

Symbolism, a 2 is read " a square," or " a to the second 
power," and means a a ; a 3 is read " a cube," or " a to the 
third power," and means a a a ; a 4 is read " a to the 
fourth power," and means a a f a* a; and, in general, a n 
is read " a to the nih power," and means a a (n times). 

This symbolism and the notion of power have been 
extended, thus : v a* is obtained by dividing a 3 by a, so 

a 2 

a 1 is defined as or a, and is read " a to the first power," 
a 

a 1 

and a is defined as or 1, and is read " a to the zero 
a 



INVOLUTION. 33 

a 1 

power," and a~ l is denned as or > and is read " a to the 

a a 

a~ l 1 

minus first power," and a~ 2 is defined as or > and is 

a as 

read " a to the minus second power/' and, in general, a~ n 
is defined as > and is read " a to the minus nth power." 

A further extension has also been made to include frac- 
tional powers, thus : 

V a 2 = V& 4 , and a i = Va 2 , so a* is defined to mean Va, 
and is read " a to the -J- power " or " the square root of a " ; 

1 n, 2 

and,' in general, a n is defined as v, and a as the mth 

power of the ^th root of a. Thus, a 1 - 25 means the 125th 
power of the 100th root of a. 

Raising numbers to high 2 4 = 16 

powers. It occasionally 2 4 = 16 

becomes necessary to raise 2 8 = 256 

a number like 2 to some ?!_ 256 

high power, as in the case 2 16 = 65536 

of 2 30 . Here the computer 2^ = 65536 

should recall the fact that 2J = 4294967296 

a m -a n = a m+n , and proceed 2 30 = 2 32 -=- 2 2 = 1073741824 
as indicated in the annexed 
multiplication. 

Powers of binomials. In the extraction of roots by the 
method to be considered it is necessary to know the cor- 
responding powers of binomials. The student may expand 
the following : 



=/ 3 + 3/% + 3/n 2 + n. 
(/+ n) 4 = (?). Obtain it by squaring (/ + n) 2 . 



Obtain it by squaring (/ + n). 



34 HIGHER ARITHMETIC. 

Exercises. 1. Square 41 by using the formula for (/+ n) 2 . 

2. Cube 22 by using the formula for (/ + n) 3 . 

3. Similarly, find the values of II 2 , II 3 , II 4 , and from the results 
find the values of II 5 and II 7 by single multiplications. 

4. Express 2~ 4 , 4" 1 , 5~ 2 , and 10~ as decimal fractions. 

5. Express 0.04 and 0.03125 as negative powers of integers. 

6. Prove that no number ending in 2, 3, 7, or 8 can be a perfect 
square. 

7. Prove that the square of a number ending in 5 ends in 025, 225, 
or 625. 

8. Prove that a square must end in 0, 1, 4, 5, 6, or 9. 

9. Prove that a cube may end in any of the digits. 

10. Prove that the cube of a number ending in 5 must end in 125, 
375, 625, or 875. 

11. Prove that the 5th power of a number ends in the same digit as 
the number itself. 

12. If the student has taken the subject of imaginary numbers in 
algebra, but not otherwise, he may solve the following : 

(a) (- i + i V^3) 2 = ? (6) (- i - i V^3) 2 = ? 

(c) (-i + W-3) 3 = ? (d) (-i-iV-3) = ? 

II. SQUARE ROOT. 

The square root of a perfect second power is one of the 
two equal factors of that power. 

In case a number is easily factored, the square root may be found 
by this means, as explained on p. 37. 

A number which is not a perfect second power has not 
two equal factors. It is, however, said to have a square 
root to any required degree of approximation. Thus, the 
square root of n to 0.1 is that number of tenths whose 
square differs from n by less than the square of any other 
number of tenths. 

E.g., the square root of 2 to 0.1 is 1.4, to 0.01 is 1.41, etc. 

It should be observed that under these definitions only abstract 
numbers can_ have square roots. Thus, 4 is the product of 2 and 2, 
hence 2 = \/4 ; but no number multiplied by itself equals $4, or 4 feet, 
or 4 square meters. 



SQUARE ROOT. 



35 



The general theory of this subject is best understood by 
following the solution of a problem. Suppose the square 
root of 547.56 be required. 

Let / = the /ound part of the root at any stage of the 
operation, and n the next digit to be found. 

Then ' (/+n)*=/* + 2fii + n\ 



The greatest square in 500 is 
v 20 has been found, and 
20 2 , or / 2 , subtracted, this 
147.56 must contain 2 fn+ ri 2 , 
or 2 20 n + n 2 . .-. by divid- 
ing by 2 20, or 40, n can be 
found approximately. .-. n = 3. 
.-. 2fn + w 2 , or 2-20-3 + 3 2 = 



2 3. 4 

5'47.56 

400. 

1 47.56 contains 2fn + n 2 . 



that is, the square of 20, or/ 2 . 



1 29 



/ is now 20 because that is 
all that has been found. 
n is now 3, the next digit. 

= 2fn + n 2 . 



v 23 has been found, and 
23 2 , or/ 2 (= 400 + 129), sub- 
tracted, this 18.56 must con- 
tain 2/n + w 2 , or 2 23 n + n 2 . 
.-. by dividing by 2 23, or 
46, n can be found approxi- 
mately. .-.n = 0.4. .-.2/n+n 2 , 
or2-23-0.4 + 0.4 2 



18.56 contains 2/n + n 2 . 

/ is now 23 because that 
has been found. 

n is now 0.4, the next 
digit. 



= 18.56 = 2/n + n 2 . 



The actual computation may be conveniently arranged 
in either of the following ways, the first being preferable 
for the majority of students. 



40 
43 


2 3. 4 

5'47.56 
400 


1 47 
1 29 


46 
46.4 


18.56 
18.56 



43 

46.4 



2 3. 4 

5'47.56 

1 47 
18.56 




For those who desire a complete explanation of the 
process a more extended discussion appears on p. 36. 



36 HIGHER ARITHMETIC. 





2 3. 4 






5<47.56 




/! 2 = 


4 00.00 




2/i =40 


147.56 contains 2j 


?!! + V / = 20 


2/ 1 + w 1 = 43 


1 29.00 = 


(( 


2/ 2 =46 


18.56 contains 2j 


^2 + ^2 2 /2 = 23 


2/ 2 -f ?i 2 = 46.4 


18.56 = 


" ^ 2 = 0.4 



1. v the highest order of the power is 100's, the highest order of 
the root is 10's, and it is unnecessary to look below 100's for the 
square of 10's. 

2. Similarly, it is unnecessary to look below 1's for the square of 
1's, below lOOths for the square of lOths, etc. [These places may be 
indicated by points ('), as in the above example.] 

3. The greatest square in the 100's is 400, which is the square of 
20, which may be called f\ (read u /-one"), the first found part of 
the root. 

4. Subtracting, 147.56 contains 2/n + n 2 because/ 2 has been sub- 
tracted from/ 2 + 2fn + n 2 , where /stands always for the found part 
and n for the next order of the root. 

5. 2/n, + n 2 is approximately the product of 2 /and n, and hence, 
if divided by 2/, the quotient is approximately n. .. n = 3. 

6. .-. 2/ + n 2 X 20 + 3 = 43, and this, multiplied by n, equals 
2/n + n 2 . 

7. v / 2 has already been subtracted, after subtracting 2/n + n 2 
there has been subtracted/ 2 + 2/n + n 2 , or (/+ n) 2 , or 23 2 . 

8. Calling 23 the second found part, / 2 , and noticing that 
/a =/i + i, it appears that 23 2 , or/ 2 2 , has been subtracted. 

9. .-. the remainder 18.56 contains 2/ 2 n 2 + n. 

10. Dividing by 2/ 2 for the reason already given, n 2 = 0.4. 

11. .-. 2/ 2 + n 2 = 46.4, and 18.56 = 2/ 2 n 2 + n 2 2 , as before. 

12. Similarly, the explanation repeats itself after each subtraction. 

13. Students will remember from algebra that every number has 
two square roots, one + and the other . .-. V547.56 = 23.4, but 
the positive root is the only one likely to be needed in practice. 

The subject of square root is still further discussed in 
the Appendix, Note 1. 



SQUARE ROOT. 37 

Common fractions. There are three general methods 
for extracting the square root of a common fraction. 

1. The square root of both terms may be extracted, as 
is advisable when each is a square number. 
E.g., 



2. The fraction may be reduced to the decimal form, as 
is advisable when this can easily be done. 

E.g., V^ = Vo! = 0.447 ..... 

3. The fraction may be reduced to an equal fraction 
whose denominator is a square number. 

E.g.,^ = VJf = \ Vl4 = \ of 3.741657 = 0.534522, an easier 
method in most cases than either of the two just mentioned. 

Factoring method. The cube root of a perfect third 
power is one of the three equal factors of that power, and 
similarly for the fourth, fifth, ..... nth roots. As mentioned 
on p. 34, such roots can often be found by factoring. 
E.g., 85,766,121 = 3-3-3-3-3-3-7-7-7-7-7-7; 
.-. V85,766,121 = 3-3-3-7-7-7 = 9261, 
%/85,766,121 = '3 3 7 7 = 441, 



and V85,766,121 =3-7 = 21. 

Even in case a number is not a perfect power the factoring method 
can often be advantageously used. 

E.g., V882 = V2 3 2 7 2 = 3 7 V2 = 21 V2 
= 21-1.4142 = 29.698 

Exercises. 1. The student may test his knowledge of the general 
theory by answering the following questions : (a) If you separate into 
periods of two figures each, where do you begin ? Consider, for 
example, the square root of 14.4. 

(b) Why does the remainder contain 2 fn + n 2 the first time ? the 
second ? 

(c) Why is 2 / always taken as the trial divisor ? 

(d) Why is n added to 2 / to make the complete divisor ? 

(e) In the example on p. 35, why does 129 equal 2 fn + n 2 ? 

(/) In that example, how can 129 and 18.56 each equal 2fn + n 2 ? 



38 HIGHER ARITHMETIC. 

2. Extract the square roots of the following numbers, writing out 
the solutions in the full form given on p. 35 : 

(a) 80.4609. (&) 8226.49. -(c) 1280.9241. (d) 0.21224449. 

(e) 12.8881. (/) 0.49112064. (g) 592330.3369. (h) 32.26694416. 

3. Extract the square roots of the following numbers, abridging 
the solution as in the first at the foot of p. 35 : 

(a) 40509.6129. (6) 0.501361708761. (c) 234.579856. 

(d) 96.27534400. (e) 1.47403881. (/) 416.05800625. 
(g) 28597039.6644. (h) 8260.628544. (i) 85747600. 

4. Extract the square roots, to 0.001, of the following numbers : 
(a) 0.0068. (6) 20. (c) 2. (d) 951. 

(e) 680. (/) 809. (g} 13. (h) 1000. 

5. Extract the square roots, to 0.00001, of the following numbers : 
(a) 976. (6) 887. (c) 0.565. (d) 3. 

6. Decide which of the three methods for extracting the square 
root of a common fraction is the best for each of the following num- 
bers, giving the reason, and extract the root accordingly, to 0.001. 

(a) TV ( & ) A- ( c ) I - (d) mi- 

(e) &. (/) iff*. (g) f. (h) l\. 

(0 *flk- 0') T h- (*) 50 ~ 2 - (0 M-. 

7. By separating into factors, find the square roots of 2304, 9216, 
396,900, 194,481, 11,025, 117,649. 

8. Similarly, the cube roots of 46,656, 91,125, 1,953,125, 11,390,625, 
250,047,000, 85,766,121, 1,771,561. 

9. Similarly, the fourth roots of 15,752,961, 43,046,721, 59,969,536, 
96,059,601. 

10. Similarly, the fifth roots of 59,049, 4,084,101, 9,765,625, 
3,486,784,401. 

11. Similarly, the sixth roots of 34,012,224, 113,379,904, 177,978,- 
515,625. 

12. The following sums are the squares of what numbers ? 

(a) II 2 + 60 2 . (6) 8088 2 + 1,022,105 2 . (c) 13,552 2 + 936,975 2 . 
(d) 18 2 + 19 2 + 20 2 + 21 2 + 22 2 + 23 2 + 24 2 + 25 2 + 26 2 + 27 2 + 28 2 . 

13. Draw a square whose side is/ + n (f may be taken as of an 
inch and n as J of an inch). From this figure, show that the square 
on / + n is made up of the square on /, plus the square on n, plus two 
rectangles which are / long and n wide, thus illustrating the fact that 



14. From the figure of Ex. 13, show that after / 2 is taken away 
there remains 2/n + w 2 . 



CUBE ROOT. 



39 



III. CUBE BOOT. 

The complete definition of cube root may be inferred 
from that of square root. Suppose the cube root of 
139,798,359 be required. Since the theory so closely 
resembles that of square root, the explanation is given 
in analogous form. 



5 1 9 

139,798,359 
/ 3 = 125,000,000 





3/n 


3/ 2 + 3/n 


14,798,359 contains 3/ 2 n + 3/n 2 + n 3 . 


3/ 2 


+ n 2 


+ n 2 


/i = 500. 








7,651,000 = 3/ 2 n + 3/n 2 + n 3 . 


750,000 


15,100 


765,100 


m = 10. 








7,147,359 contains 3/ 2 n + 3/n 2 + n 3 . 








/ 2 = 510. 


780,300 


13,851 


794,151 


7,147,359 = 3/ 2 n + 3/n 2 + n 3 . 








n 2 = 9. 



1. v the highest order of the power is hundred-millions, the highest 
order of the root is 100's (why ?), and it is unnecessary to look below 
millions for the cube of 100's. (Why ?) 

2. Similarly, it is unnecessary to look below 1000's for the cube of 
10's, below 1's for the cube of 1's, etc. (These periods may be indi- 
cated by points as in square root, if desired.) 

3. The greatest cube in the hundred-millions is 125,000,000, the 
cube of 500. .-. 500 may be called/. 

4. Subtracting, 14,798,359 contains 3/ 2 n + 3/n 2 + n 3 . (Why ?) 

5. This is approximately the product of 3/ 2 and n, and hence if 
divided by 3/ 2 the quotient is approximately n. .-. n = 10. 

6. .*. 3/n + ?i 2 = 15,100, and 3/ 2 + 3/n + n 2 = 765,100, and this, 
multiplied by n, equals 3/ 2 n + 3/n 2 + n 3 . 

7. v / 3 has already been subtracted, after subtracting 3/ 2 n + 3/n 2 
+ n 3 there has been subtracted (/ + n) 3 , or 510 3 . 

8. Calling 510 the second found part, / 2 , it appears that / 2 3 has 
been subtracted. .-. the remainder contains 3/ 2 n + 3/n, 2 + n 3 . 

9. The explanation now repeats itself as in square root. 



40 HIGHER ARITHMETIC. 

In practice, the work is usually arranged somewhat as 

follows : 

5 1 9 
139,798,359 
125 



7500 
7651 


14798 
7651 


780300 
794151 


7 147 359 
7 147 359 



The subject of cube root is still further discussed in the 
Appendix, Note II. 

Exercises. 1. (a) If you separate into periods of three figures 
each, why do you do so ? Where do you begin ? Why ? Consider, 
for example, the cube root of 13.31. 

(6) Why does the second remainder contain 3/ 2 w + 3/n 2 + n 3 ? 

(c) Why is 3/ 2 always taken as the trial divisor ? 

(d) Why is 3/n + n 2 added to 3/ 2 to make the complete divisor ? 

(e) In the example on p. 39, why does 7,651,000 equal 3/ 2 n + 
3/n 2 + n 3 ? 

(/) In that example, how can 7,651,000 and 7,147,359 each equal 
3/% + 3/w 2 + n 3 ? 

2. Extract the cube roots of the following numbers, writing out the 
solutions in, the full form given on p. 39 : 

(a) 139,798,359. (6) 248,858.189. (c) 0.004657463. 

(d) 19.902511. (e) 0.000091733851. (/) 731.432701. 

3. Extract the cube roots of the following numbers, abridging the 
solution as suggested in square root and at the top of this page : 

(a) 553,387,661. (6) 381.078125. (c) 997.002999. 

(d) 0.051064811. (e) 0.0001851930. (/) 0.876467493. 

4. Extract the cube roots, to 0.001, of the following numbers : 
(a) 251. (6) 455,000. (c) 0.57. (d) 0.27. 

(e) 998. (/) 0.007. (g) 0.194104601. (h) 0.47637955. 

5. Explain three methods of extracting the cube root of a common 
fraction, analogous to those given for square root. 

6. Decide which of these three methods is the best for each of the 
following numbers, giving the reason, and extract the root accord- 
ingly, to 0.001 : 



CHAPTER V. 
The Formal Solution of Problems. 



THE most important portion of arithmetic, considered 
from the business standpoint, has already been completed, 
especially essential being that part which treats of addi- 
tion, subtraction, multiplication, and division of integers 
and fractions. The subsequent portions of the subject are 
taught for the business and scientific principles involved, 
but largely as an exercise in logic. And since the opera- 
tions mentioned have been the subject of extensive drill in 
the lower grades, it is neither necessary nor advisable to 
preserve them, after checking the result of each computa- 
tion, in the treatment of applied problems. The solution 
should now be logically arranged in steps numbered for 
reference, the complete operations being preserved when- 
ever the teacher advises. In this way, the logic of the 
solution stands out prominently, while on the other hand 
there is no loss in the way of arithmetical computations. 

At every stage of the solution time and energy should 
be economized by resort to factoring and cancellation. It 
is a good rule, never multiply till you have to, always 
factor if you can. The advantages of this rule are seen in 
the problem solved on p. 47. 

The student should also be advised as to the proper use 
of symbols and language, and to this end a few suggestions 
may be of value. 



42 HIGHER ARITHMETIC. 



I. SYMBOLS. 

The common symbols for multiplication are X and , 
the latter being preferable for students sufficiently mature 
not to confuse it with the decimal point. It is advisable 
to write the multiplier first because (a) it is usually read 
first, (&) the tendency among leading writers is to place it 
first, and (c) in an algebraic expression like 4 x the first 
factor is usually looked upon as the multiplier. 

Thus, if 1 book costs $2, 123 books, at the same rate, will cost 
123 $2. This would be indicated by the step 123 $2 = $246, but 
the actual multiplication would of course be 2 123, on the principle 
that 123 2 $1 = 2 123 $1. 

The symbols may therefore be read as follows : 

2 $3, or 2 X $3, "2 times $3," or " 2 into $3 " ; 
$3 2, or $3 X 2, " $3 multiplied by 2." 

The word " times " in this connection has a much broader meaning 
than that assigned when arithmetic was in its infancy. Thus, we 
say "2 times $4," meaning thereby "2 times $4 and $ of $4." It 
is not customary, however, to use the word after a proper fraction ; 
thus, &-$4 is read " of $4," and ft. is read "f of a foot." 
Hence, 2 times $4 has acquired a meaning; but to look out of the 
window 2 times is nonsense. 

The general agreements of mathematicians as to the 
relative weight of symbols should also be understood. The 
usage varies in different countries, however, and occasion- 
ally is not entirely settled in any one. The following may 
be taken as indicating the rules followed by the leading 
writers of the day. 

The absence of a sign between two letters, and the frac- 
tional notation, indicate operations to be performed before 
any others. 

Thus, in the expression a -r- cd -f- 'the multiplication cd is first 

y 

performed ; then the division of e by g ; then the other divisions in 
order, beginning at the left. 



THE FORMAL SOLUTION OF PROBLEMS. 43 

The word " of " following a fraction stands next as to 
weight. 

Thus, in a -f- ^ of cd, the multiplication cd is first performed ; then 
the multiplication by % ; then the division of a by the result. 

The symbols , X, -f-, / stand next, one having the same 
weight as another. 

Thus, inl-3x4 + 2X 6/3, the operations are performed in order 
from left to right, the result being 12. 

The symbols +, stand next, one having the same 
weight as the other. 

Thus, 2 + 3-6-5 + 4-8-|- = 35. 

The symbol : , when used as a symbol of ratio, stands 
next. 

Thus, 2 + 3:4 + 1 = 5:5 = 1. But since the symbol is one of 
division, and in some countries is the leading symbol of that opera- 
tion, it is frequently given the same weight as the -r. In that case, 
2 + 3 : 4 + 1 = 3f , while (2 + 3) : (4 + 1) = 1. 

The other common symbols are sufficiently understood 
already, or are explained elsewhere in this work. 

HISTORICAL NOTE. The symbols + and were used by Wid- 
inann in an arithmetic published at Leipzig in 1489, = by Recorde 
in an algebra published in 1557, X by Oughtred in 1631, the dot () 
as a symbol of multiplication by Harriot in 1631, the absence of a 
sign between two letters to indicate multiplication by Stifel in 1544, 
: as a symbol of division by Leibnitz, + as a symbol of division by 
Rahn in an algebra published at Zurich in 1659, > and < by Harriot 
in 1631. The symbols =56, ^>, <, indicating "not equal," etc., are 
recent. Parentheses were first used as symbols of aggregation by 
Girard in 1629. The decimal point came into use in the seventeenth 
century ; it seems to have appeared first in a work published by 
Pitiscus in 1612, but it was not extensively employed until more than 
a century later. Positive integral exponents in the present form were 
first used by Chuquet in 1484. The symbol V~ was first used in this 
form by Rudolf! in 1525. 



44 HIGHER ARITHMETIC. 



II. LANGUAGE. 

The student should also guard against statements like the 
following : " 2 times greater than 3 " for " 2 times as great 
as 3"; "3 + 1 equals to " "$4 + 3" for $4 + $3" ; 
2 X 3 = $6 " for " 2 X $3 = $6 " ; "2 is contained in 
$6 $3 times " for $6 divided by 2 equals $3," or " of 
$6 equals $3." 

III. METHODS. 

There is no general method of solution covering all prob- 
lems ; if there were, the subject would lose substantially 
all of its value as an exercise in logic. The student should 
feel encouraged to put into the work all the individuality 
possible, only being sure (1) that each statement is true, 
(2) that each result is checked, (3) that his solution involves 
no undue labor. 

1. Analysis in general. The solution of any problem of 
applied arithmetic requires analysis of some kind ; in other 
words, the application of a student's common sense. Two 
types are here given, and it will be seen that if the steps 
are properly arranged the oral analysis is a simple matter, 
beginning at each stage with a " since " and reasoning to a 
" therefore,' 7 

Problem. If the average velocity of a bullet in going from a gun 
to a target is 1342 ft. per sec., and that of sound is 1122 ft. per sec., 
how much time will elapse, on a range of 1000 yds., between the time 
the bullet strikes the target and the time that the sound of the dis- 
charge reaches the target ? 

Solution. 1. 1000 3 ft. = 3000 ft. 

2. The bullet goes 1 ft. in y^ sec. 

3. .-. it goes 3000 ft. in ffff sec., or 2.24 sees. 

4. Similarly, sound requires ff f sec., or 2.67 sees. 

5. 2.67 sees. 2.24 sees. = 0.43 sec. 



THE FORMAL SOLUTION OF PROBLEMS. 45 

Analysis, v 1 yd. = 3 ft., .-. 1000 yds. = 1000 3 ft. 
v the bullet goes 1342 ft. in 1 sec., .-. it goes 1 ft. in yJ^ of 1 sec., 
and 3000 ft. in 3000 T ^ 7 of 1 sec. 

Similarly, sound goes 3000 ft. in 3000 T1 ^j of 1 sec. 
The difference in time is evidently the result required. 

Problem. At what time between 1 and 2 o'clock are the hands of 
a clock at right angles to each other ? 

Analysis. (The student should first draw the figure.) 

v they are together at 12 it is readily seen that the minute hand 
must gain 60 minute-spaces on the hour hand to bring them together 
again. 

.-. it must gain (60 + 15) minute-spaces or else (60 + 45) minute- 
spaces to bring them at right angles between 1 and 2. 

v at 1 o'clock the minute hand is at 12 and the hour hand at 1. 

.-. the former gains 55 minute-spaces in 1 hr., or 1 minute-space in 

& hr - 

.-. to gain 75 or 105 minute-spaces it requires 75 -fa hr. = 1 hr. 

21 r 9 T mins., or 105 -fa hr. = 1 hr. 54 T 6 T mins. 

Exercises. 1. What is the speed in feet per sec. of a train moving 
uniformly at the rate of 20 mi. per hr. ? 60 mi. per hr. ? 

2. The earth's center moves about the sun at the average rate of 
101,090 ft. per sec.; how many miles per hr.? 

3. An elastic ball rebounds to a height which is f of that through 
which it fell ; on the third rebound it rises to a height of T 4 r ft. ; from 
what height did it first fall ? 

4. Each of the two arctic zones covers 0.02 of the earth's surface, 
and each of the temperate zones 0.26 ; what part is covered by the 
two torrid zones together ? 

5. A locomotive consumes ^ of its tankful of water every mile ; it 
starts with only f of a tankful ; how many miles has it gone when it 
has T 2 5 of a tankful left ? 

6. A steam engine using 28.5 tons of coal in 30 working days has 
an improvement effected rendering it necessary to use only 4.8 tons a 
week of 6 working days ; how much is saved in a year of 300 working 
days, coal costing $5.60 a ton, not considering the cost of the improve- 
ment ? 

7. If the pressure of air on the surface of a lake is 15 Ibs. per 
sq. in., and if 1 cu. ft. of water weighs 1000 oz., find the pressure per 
sq. ft. at the depth of 100 ft. 



46 HIGHER ARITHMETIC, 

8. The average daily motion of the earth about the sun is 59' 8.3" 
a day ; that of Mars is 31' 26. 5" ; if they moved in the same plane 
and kept these rates, how many days would elapse from the time they 
were in the same straight line on the same side of the sun to the time 
when the earth was again directly between Mars and the sun ? Draw 
a diagram illustrating the problem. 

9. Similarly for earth and Mercury, the latter's daily rate being 
4 5' 32.5". 

10. Similarly for earth and Venus, the latter's daily rate being 
1 36' 1.1". 

11. Similarly for earth and Neptune, the latter's daily rate being 
21.5". 

12. Similarly for Mercury and Venus (see Exs. 9, 10). 

13. Similarly for Venus and Neptune (see Exs. 10, 11). 

14. What are the relative positions of the earth, the moon, and the 
sun at the time of a new moon ? The average daily motion of the 
moon about the earth is 13. 1764 ; the apparent daily motion of the sun 
in the same direction is 0.98565 ; required the time from one new 
moon to another. 

15. It is estimated that a cannon-ball leaving the earth at the rate 
of 500 mi. per hr., and continuing that rate to the nearest fixed star, 
would require about 4,500,000 yrs. for the journey, (a) Express in 
index notation, giving only the first two significant figures, the dis- 
tance to the star. (6) Knowing that light travels about 186,000 mi. 
per sec., find, to 0.1, the number of years required for the light of the 
star to reach the earth. (Take 365J da. = 1 yr.) 

16. Sound travels 65,400 ft. per min.; how far away is a gun 
whose report is heard 15 sees, after firing ? 

17. The average cost per day for tuition in the common schools of 
the United States is 8.2 cts. Estimating the number of pupils at 
14,000,000, what is the total cost per day ? What is the cost per year 
of 150 school days ? 

18. The velocity of light being 186,330 mi. per sec., how long does 
it take the light from the sun to reach the earth, the distance being 
93, 165,000 mi.? 

19. A factory is insured for $2500 in one company, $3500 in 
another, and $2000 in another ; it is damaged by fire to the extent 
of $4875 ; what portion of the loss should each company bear ? 

20. A man left by will to four persons the sums of $1000, $950, 
$800, $750, respectively ; his estate produced only $2900 ; how much 
should each legatee receive ? 



THE FORMAL SOLUTION OF PROBLEMS. 47 

2. Unitary analysis is so called because the student 
analyzes the problem by passing to one or more units. 
The method is very advantageous in the solution of many 
problems which, although of no especial value in business 
or in science, are still found in most text-books. While 
such problems are foreign .to the spirit of the present 
work, a few are given by way of illustration. In all these 
cases the words " at the same rate " are to be understood. 

Problem. How many pumps working 12 hrs. per da. will be 
required to raise 7560 tons of water in 14 da., if 15 pumps working 
8 hrs. per da. can raise 1260 tons in half that number of days ? 

Solution. 1. 1260 t. raised in 7 da. of 8 hrs. each require 15 times 
the work of 1 pump. 

2. 1 t. in 7 da. of 8 hrs. requires T ^Q times the work of 1 pump. 

3. 1 t. in 1 da. of 8 hrs. " 7 T jfo " 

4. 1 t. in 1 da. of 1 hr. " 8 7 T ^ " 

5. 7560 t. in 1 da. of 1 hr. " 7560 8 7 T |f^ " 

6. 7560 1. in 14 da. of 1 hr. " T ^ 7560 8 7 T || - " 

7. 7560 t. raised in 14 da. of 12 hrs. each require 

Ta ' i? ' 756 ' 8 7 T if o times the work of 1 pump 
= the work of 30 pumps. 

In actual practice it would be better to pass from step 1 to step 4, 
and then directly to step 7. It would be a waste of energy to per- 
form the operations at each step; by waiting until the last step 
numerous cancellations simplify the computation. 

Exercises. 1. If 10 yds. of cloth yd. wide cost $6.25, how 
much will 15 yds. of that cloth 1 yd. wide cost at the same rate per 
sq. yd. ? 

2. How long will it take 12 men to do a piece of work which 8 men 
can do in 54 da. ? 

3. If a 5-ct. loaf of bread weighs 1.5 Ibs. when wheat is 75 cts. per 
bu., what should it weigh when wheat is $1.00 per bu.? 

4. If 5 compositors in 16 da. of 10 hrs. each can set up 20 sheets 
of 24 pages each, 40 lines to a page and 40 letters to a line, in how 
many days of 8 hrs. each can 10 compositors set up a volume com- 
posed of 40 sheets of 16 pages to the sheet, 60 lines to a page and 50 
letters to a line ? 



48 HIGHER ARITHMETIC. 

3. The simple equation. Few mathematicians now assert 
that the distinction between arithmetic and algebra lies in 
the use of letters as symbols of quantity. It is impossible 
to study exhaustively the science of number without using 
literal notation. The simple equation is now used in all of 
the grades of many grammar schools, and it is no innova- 
tion to suggest its use in a work of this nature. 

No more difficult equation is necessary than one of the 
following type : 

1. Given ax + b = c, to find x. 

2. ax = c 6, by subtracting 6 from these equals. Ax. 3 

3. x = by dividing these equals by a. Ax. 7 

4. Check : Putting - - f or x in step 1, a h b = c. 

a a/ 

A single illustration may be given : 

What sum gaining 0.06J of itself in a year amounts to $157.50 in 
2 yrs. ? 

1. Let x = the sum. 

2. 2 O.OGi = 0.12|. 

3. .-. x + 0.12z = $157.50. 

4. .-. 1.12ix = $157.50. 

5. .-. x = $140. 

Exercises. 1. What number is that which divided by 17 equals 
2.1? 

2. Divide 10 into two parts such that twice one part equals 3 times 
the other. 

3. A fulcrum is to be placed under a 3-ft. lever so as to divide it 
into two parts such that 1.2 times the first shall equal 4.8 times the 
second ; how far is it from either end ? 

4. Alcohol as received in the laboratory is 0.95 pure ; how much 
water must be added to a gallon of this alcohol so that the mixture 
shall be half pure ? 

5. Air is composed of 21 volumes of oxygen and 79 volumes of 
nitrogen ; if the oxygen is 1.1026 times as heavy as air, the nitrogen 
is what part as heavy as air ? 



CHECKS. 49 



IV. CHECKS. 

A good computer checks his work at every step, and the 
student who does this has no need of the printed answers 
to problems involving only numerical calculations. 

Checks have already been given for the fundamental 
operations, the most valuable one being that of casting out 
nines. One other deserves especial mention in tins con- 
nection : Always form a rough estimate of the answer before 
beginning a solution. Beach as close an approximation as 
possible in a short time. This will be found to check most 
large errors and to do away with the absurd results often 
given by careless students. 

E.g., if the interest on $475 for 1 yr. at 4|% is required, the 
student should at once think that it is a little less than half the inter- 
est on $1000, that is, a little less than half of $45 ; he might therefore 
make the estimate $20. The interest is really $21.38. 

In the following exercises form a rough estimate of the 
answers and write down the approximations. Then solve. 

Exercises. 1. At 12| cts. a pound, how much will 2f Ibs. of 
cheese cost ? (In solving note that 12 = J-f^.) 

2. At 37| cts. a yard, how much will 13 yds. of cloth cost ? (In 
solving note that 37| f of 100.) 

3. At $3.50 a barrel, how much will 68 bbls. of flour cost ? 

4. At $1.70 a barrel, how much will 126 bbls. of apples cost ? 

5. At 45 cts. a yard, how many yards of cloth can be bought for 
$6.75 ? 

6. If a person's taxes are 5.8 mills on $1, how much will they be 
on $8500 ? 

7. If 41 qts. of water weigh as much as 51 qts. of alcohol, and 1 qt. 
of water weighs 2.2 Ibs., how much will 1 qt. of alcohol weigh ? 

8. Bronze contains by weight 91 parts of copper, 6 of zinc, and 3 
of tin ; how many pounds of each in 700 Ibs. of bronze ? 

9. In drawing a picture of a tower which is 160 ft. high and 35 ft. 
in diameter, the diameter is to be represented by 5 in. ; by how many 
inches should the height be represented ? 



50 HIGHER ARITHMETIC. 

Exercises. 1. How much water must be added to a 5% solution 
of a certain medicine to reduce it to a 1% solution ? 

2. If sound travels 5450 ft. in 5 sees, when the temperature is 32, 
and if the velocity increases 1 ft. per sec. for every degree that the 
temperature increases above 32, how far does sound travel in 8 sees, 
when the temperature is 70? 

3. 84J qts. of water are drawn through a pipe every 4 mins. from 
a tank containing 237 qts. ; how many minutes will it take to empty 
the tank, supposing the water to continue to run at the same rate ? 

4. The total debt of the United States government Jan. 1, 1897, 
was about $1,785,412,641, and the estimated population on that day 
was 74,036,761 ; what was the debt per capita ? 

5. A clock is set on Monday at 7 A.M.; on Tuesday at 1 P.M., cor- 
rect time, it is 3 mins. slow ; how many minutes will it be behind at 
7 A.M., correct time, on Saturday ? 

6. If a railroad charges $18 for transporting 12,000 Ibs. of goods 
360 mi., how much ought to be charged for transporting 15,000 Ibs. of 
goods 280 mi. at the same rate ? 

7. If in. on a map corresponds to 7 mi. of a country, what dis- 
tance on the map represents 20 mi. ? 

8. How much pure alcohol must be added to a mixture of alcohol 
and $ water, so that ^ of the mixture shall be pure alcohol ? 

9. If 40 pupils use 6 boxes of crayons, 200 in a box, hi 3 mo., 
how many boxes, 150 in a box, will be required, at the same rate, to 
supply 75 pupils for 2 mo. ? 

10. If the velocity of electricity is 288,000 mi. per sec., how long 
will it take electricity to travel around the earth, 24,900 mi. ? 

11. The respective rates per sec. at which sound travels through 
air, water, and earth are approximately 1130 ft., 4700 ft., and 7000 ft. ; 
at these rates, in what time could sound be transmitted a distance of 
6 mi. through each of these media ? (1 mi. = 5280 ft.) 

12. A certain sum of money gains of itself, the total amount then 
being $728 ; what is the sum gained ? 

13. One of the trains on the Caledonian railway from Carlisle to 
Stirling, 117f mi., makes the run in 124 mins.; the "Empire State 
Express" makes the run from Syracuse to Rochester, 80 mi., in 84 
mins. ; what is the average rate of each per hr. ? 

14. The total debt of the various states and territories at the time 
of the eleventh census was $1,135,210,442, which was $18.12 per 
capita; compute the total population at that time, correct to 1000. 



CHAPTER VI. 
Measures. 



THE earlier business arithmetics contained a large num- 
ber of tables of measures, a necessity when the world was 
divided into relatively small states, each with its own 
system of coinage, weights, etc. As an example of the 
number of tables in use in a single country, there were 
nearly four hundred ways of measuring land in France at 
the close of the eighteenth century. Moreover, certain 
trades adopted special measures, thus adding to the con- 
fusion. The result is seen in the tables of Troy, avoirdu- 
pois and apothecary weights, the wine, beer, apothecary 
and common measures of capacity, besides numerous special 
units practically obsolete in general business in America, 
as the stone, long hundredweight, etc. 

For common use to-day, only a few tables are needed. 
If one is to enter some trade which continues to use 
special measures, as that of druggist, the tables should 
be learned at that time as part of his technical education. 
Similarly, in an exchange office one must learn a consider- 
able number of money systems, but for general information 
three or four suffice. The problems here set for review 
require only those tables in general use in business or in 
the sciences. 

The tables on pp. 52 and 53 are inserted chiefly for 
reference. They include those which the student will most 
need to review. The metric tables are given on pp. 61 
and 62. 



52 



HIGHER ARITHMETIC. 



TABLES OF COMMON MEASURE NEEDING REVIEW. 



COUNTING BY 12. 
1 dozen (doz.) = 12. 

1 gross (gro.) = 12 2 . 

1 great gross (gt. gro.) = 12 3 . 

COUNTING SHEETS OF PAPER. 
24 sheets = 1 quire. 

20 quires, or 480 sheets = 1 ream. 

COMMON MEASURES OF LENGTH. 

12 inches (in.) = 1 foot (ft.). 

3 feet = 1 yard (yd.). 

6i yards, or 16|ft.= 1 rod (rd.). 

320 rods, or 5280 ft. = 1 mile (mi. ). 

SURVEYORS' MEASURES OF LENGTH. 
7.92 inches = 1 link (li.). 
100 links = 1 chain (ch.). 
80 chains = 1 mile. 



MISCELLANEOUS MEASURES 
OF LENGTH. 

4 inches = 1 hand. 

6 feet = 1 fathom. 

1.15 miles, nearly, = 1 knot, or 

1 nautical or geographical mile. 

LIQUID MEASURE. CAPACITY. 
4 gills (gi.) = 1 pint (pt.). 

2 pints = 1 quart (qt.). 
4 quarts = 1 gallon (gal.) 

= 231 cubic inches. 
Barrels and hogsheads vary in 
size. 

DRY MEASURE. CAPACITY. 
2 pints = 1 quart. 
8 quarts = 1 peck (pk.). 
4 pecks = 1 bushel (bu.) 
= 21 50.42' cu. in. 



AVOIRDUPOIS WEIGHT. 
16 ounces (oz.) = 1 pound (lb.). 
100 pounds = 1 hundredweight (cwt.). 
2000 pounds = 1 ton (t.). 



2240 pounds = 1 long ton, little used in America except in whole- 
sale transactions in mining products, and not generally there. 



COMMON MEASURES OF SURFACE. 

144 square inches (sq. in.) = 1 square foot (sq. ft.). 

9 square feet = 

30 J square yards = 
160 square rods 

640 acres = 

1 mile square = 



square yard (sq. yd.). 

square rod (sq. rd.). 

acre (A.). 

square mile (sq. mi.). 

section. 



36 square miles 



= 1 township. 



100 square feet = 1 square (of roofs, etc.). 



MEASURES. 53 



CUBIC MEASUKE. 

1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.). 
27 cubic feet = 1 cubic yard (cu. yd.). 



128 cubic feet = 1 cord. The word "cord" is generally used, 
however, to mean a pile of wood 8 ft. long and 4 ft. high, the price 
depending (other things being the same) on the length of the stick. 

1 cubic yard = 1 load (of earth, etc.). 
24f cubic feet = 1 perch. 

ENGLISH MONEY. 

12 pence (d.) = 1 shilling (s.) = $0.243 +. 
20 shillings = 1 pound () - $4.8665. 

FRENCH MONEY. 

100 centimes = 1 franc (fr.) =.$0.193. 

The French system is also used in several other countries, as in 
Belgium, Switzerland, Italy, etc., but the names are not uniform, in 
Italy, for example, the franc being called a lira. 

GERMAN MONEY. 
100 pfennigs = 1 mark (M.) = $0.238. 

APOTHECARIES' WEIGHT. 
20 grains (gr.) = 1 scruple (sc. or 9). 
3 scruples = 1 dram (dr. or 5)- 
8 drams = 1 ounce (oz. or ). 
12 ounces = 1 pound (lb.). 
6760 grains = 1 pound. 

The table of apothecaries' weight is used in selling drugs at retail. 

TROY WEIGHT. 

24 grains (gr.) = 1 pennyweight (pwt. or dwt.). 
20 pennyweights = 1 Troy ounce. 
12 Troy ounces = 1 Troy pound. 
437.5 grains = 1 Avoirdupois oz. 

7000 grains = 1 Avoirdupois lb. 

480 grains = 1 Troy oz. 

5760 grains = 1 Troy lb. 

Troy weight is used for precious metals. 



54 HIGHER ARITHMETIC. 

I. COMPOUND NUMBERS. 

When a concrete number is expressed in several denomi- 
nations it is called a compound number. 

E.g., 3 ft. 6 in. But 3.5 ft. and -$2.25 are not compound numbers. 

Reduction of compound numbers is a process so familiar 
to the student that two examples will satisfy for illustra- 
tion. 

(1) Reduction descending. Eeduce 365 da. 5 hrs. 48 mins. 
to minutes. 

Explanation and solution. 1. v 1 da. = 24 hrs. 

2. .-. 365 da. = 365 X 24 hrs. = 8760 hrs. 

3. 8760 hrs. + 5 hrs. = 8765 hrs. 

4. v 1 hr. = 60 mins. 

5. .-. 8765 hrs. = 8765 X 60 min. = 525,900 mins. 

6. 525,900 mins. + 48 mins. = 525,948 mins. 

Practical calculation. (The notes in parentheses explain the opera- 
tions.) 

365 

(Multiply by 3)= 1095 (For 24 = 3 X 8.) 
(Multiply by 8 and add 5)= 8765 (i.e., 365 X 24 hrs. + 5 hrs.) 
(Multiply by 60 and add 48)= 525,948 (i.e., 8765 X 60 mins. +48 mins.) 

(2) Reduction ascending. Eeduce 525,948 mins. to days, 
etc. 

Explanation and solution. 1. v 1 min. = ^ hr. 

2. .-. 525,948 mins. = 525,948 X ^ hr. = 8765 hrs. and ff hr. or 
48 min. 

3. v 1 hr. = J da. 

4. .-. 8765 hrs. = 8765 X J da. = 365 da. and ^ da., or 5 hr. 

5. .-. 365 da. 5 hrs. 48 mins. 

Practical calculation. (The notes in parentheses explain the opera- 
tions.) 

60|525948 

24 1 8765 48 (525,948 X ^ hr.) 
365 5 (8765 X ^ da.) 



COMPOUND NUMBERS. 55 

Exercises. 1. Eeduce 43 wks. 5 hrs. 49 mins. 57 sees, to seconds. 

2. Keduce 4,568,657 sees, to weeks, days, etc. 

3. Reduce 625 cu. yds. 19 cu. ft. 1609 cu. in. to cubic inches. 

4. Reduce 1,847,638 ft. to miles, yards, and feet. 

5. Reduce 25 mi. 459 yds. 31 in. to inches. 

6. Reduce 12,563,257 sq. in. to acres, square yards, etc. 

7. Reduce 5 gals. 3 pts. to pints. 
. 8. Reduce 341 qts. to gallons. 

9. Reduce 150 Ibs. to ounces. (Avoirdupois.) 

10. Reduce 274 oz. to pounds and ounces ; to pounds and decimals 
of a pound. (Avoirdupois.) 

11. Reduce 36 gt. gro. to gross; to units. 

12. Reduce 15 reams to quires ; to sheets. 

13. Reduce 142,872 sheets to quires ; to reams. 

14. Reduce 19,436 cu. ft. to cubic yards. 

15. Reduce 2 wks. 2 da. 19.2 hrs. to the fraction of a month of 
4 wks. 

16. Reduce 14 hrs. 15 mins. to the fraction of 3 da. 

17. Reduce T 8 T pt. to the fraction of a gallon. 

18. Reduce 3 qts. 1 pt. to the decimal of a gallon. 

19. Reduce 18 hrs. 30 mins. 30 sees, to the fraction of a week. 

Compound addition and subtraction differ so little in 
theory from the addition and subtraction 

of simple abstract numbers as to require 9 Ibs. 15 oz. 

no extended review. The cases arising 10 12 

in actual practice rarely involve more _8 6 

than two denominations, the tendency 29 1 
being to reduce the lower units to deci- 
mals of the higher. Thus, while it was 29 Ibs. 1 oz. 

formerly not unusual to add numbers 10 12 

like 27 rds. 5 yds. 2 ft. 11 in., it is now 18 5 
more common to deal with numbers like 
463.4 ft., 1.27 A., 4.345 mi., etc. 

In the annexed example in addition the computer should say, "6, 
18, 33, 1 ; 2, 10, 20, 29." In the example in subtraction he should 
proceed as with simple numbers, remembering that 16 oz. = 1 lb., and 
should say, " 12 and 5 are 17 ; 11 and 18 are 29." 



56 HIGHER ARITHMETIC. 

Exercises. 1. What check should be used in compound addi- 
tion ? in compound subtraction ? 

2. Add 9 Ibs. 7 oz., 52 Ibs. 6 oz., 91 Ibs. 12 oz., 7 Ibs., 5 Ibs. 2 oz., 
13 oz. (Avoirdupois.) 

3. Add 13 t. 450 Ibs., 12 t. 700 Ibs., 342 t., 44 1. 1500 Ibs., 1200 Ibs. 

4. Add 25 gals. 3 qts., 47 gals. 2 qts. 1 pt., 15 gals. 1 qt., 1 pt., 
9 gals. 

5. Add 5 yds. 2 ft., 6 yds. 1 ft. 7 in., 9 yds. 2 ft. 5 in., 1 ft. 

6. Add 6 bu. 3 pks., 9 bu. 2 pks., 5 bu. 1 pk., 3 pks. 

7. Add 10 da. 5 hrs. 42 mins. 7 sees., 23 hrs. 10 mins. 2 sees., 11 da. 
4 mins., 5 hrs. 4 mins. 5 sees., 1 da. 15 sees. 

8. Add 7 mo. 15 da., 5 mo. 14 hrs. 3 sees., 1 mo. 7 da. 5 mins. 

57 sees., 2 mo. 54 mins., 9 hrs., 7 da., 2 mo. 

9. Add 12 mi. 3 rds. 2 ft., 3 mi. 75 rds. 10 ft., 4 mi. 12 ft., 3 rds. 
6ft. 

10. From 13 Ibs. 9 oz. subtract 9 Ibs. 10 oz. (Avoirdupois.) 

11. From 7 mo. 9 da. subtract 5 mo. 15 da. 

12. From 25 gals. 2 qts. subtract 10 gals. 3 qts. 1 pt. 

13. From 5 yds. 1 ft. 7 in. subtract 2 yds. 2 ft. 10 in. 

14. From 6 bu. 2 pks. subtract 5 bu. 3 pks. 

15. From 7 da. 5 hrs. 27 mins. 42 sees, subtract 5 da. 5 hrs. 27 mins. 
43 sees. 

16. From 87 cu. yds. 8 cu. ft. 924 cu. in. subtract 35 cu. yds. 23 
cu. ft. 1688 cu. in. 

Compound multiplication. The remarks already made 
concerning practical problems in addition and subtraction 
apply with equal force to multiplication and division. The 
general theory is evident from the analysis of the following 
problem. 

Required the product of 10 X 3 Ibs. 4 oz. CALCULATION. 

Analysis. 1. 10 X 4 oz. = 40 oz. 2 Ibs. 8 oz. 3 Ibs. 4 oz. 

2. 10 X 3 Ibs. = 30 Ibs. 10 

3. 30 Ibs. + 2 Ibs. 8 oz. = 32 Ibs. 8 oz. 32 Ibs. 8 oz. 

Compound division. The definition of division, already 
given on p. 29, should now be recalled for the purpose of 
distinguishing between the two general cases. 



COMPOUND NUMBERS. 57 

Division is the operation of finding one of two factors, 
the quotient, when the product and the other factor are 
given. 

Hence, there are two general cases illustrated by the following 
example. 

Since $10 is the product of the factors 2 and $5, 

1. .-. $10 -T- $5 = 2, the idea of measuring, being contained in, con- 
tinued subtraction. That is, $5 is contained in $10 2 times. 

2. $10 -r 2 = $5, the idea of separation, partition, multiplication 
by a fraction. That is, $10 divided by 2 equals $5, $10 has been 
separated into two parts. 

1. When dividend and divisor are compound numbers of 
the same kind, they may be reduced to the same denomination 
and the division performed in the ordinary way. 

For example, how many times does 32 Ibs. 8 oz. contain 3 Ibs. 4 oz. ? 
In this case it is more simple to reduce to pounds, thus : 

1. 32 Ibs. 8 oz. = 32.5 Ibs. 

2. 3 Ibs. 4 oz. = 3.25 Ibs. 

3. 32.5 Ibs. -r 3.25 Ibs. = 10. 

But it is usually easier to reduce to one of the lower denominations, 
especially where more than two denominations are involved. For 
example, how many times does 29 t. 87 Ibs. 2 oz. contain 3 t. 4 cwt. 
54 Ibs. 2 oz. ? 

1. 29 t. 87 Ibs. 2 oz. = 58087.125 Ibs. 

2. 3 t. 4 cwt. 54 Ibs. 2 oz. = 6454.125 Ibs. 

3. 58087.125 Ibs. -f 6454.125 Ibs. = 9. 

II. When the divisor is an abstract number. 
For example, divide 29 t. 87 Ibs. 2 oz. by 9. 

CALCULATION. 
9 ) 29 t. 87 Ibs. 2 oz. 

3 t. 454 Ibs. 2 oz. 
Analysis. 1. 29 t. -J- 9 = 3 t., and 2 t., or 4000 Ibs., remainder. 

2. 4000 Ibs. + 87 Ibs. = 4087 Ibs. 

3. 4087 Ibs. -r 9 = 454 Ibs., and 1 lb., or 16 oz., remainder. 

4. 16 oz. + 2 oz. = 18 oz. 

5. 18 oz. -r 9 = 2 oz. 

6. .-. 3 t. 454 Ibs. 2 oz. 



58 HIGHER ARITHMETIC. 

Exercises. 1. Multiply 27 gals. 3 qte. 1 pt. 3 gi. by 36, checking 
by division. 

2. Also by 236, checking by division. 

3. Multiply 17 wks. 4 da. 13 hrs. 27 mins. 36 sees, by 9, checking 
by division. 

4. Also by 79, checking by division. 

5. Multiply 23 cu. yds. 6 cu. ft. 459 cu. in. by 8, checking by 
division. 

6. Multiply the result in Ex. 5 by 9, checking by division. 

7. Multiply 512 rds. 2 yds. 2 ft. 2 in. by 6, checking by division. 

8. Multiply 2 sq. yds. 3 sq. ft. 9 sq. in. by 10, checking by 
division. 

9. Divide 878 wks. 4 da. 15 hrs. 37 mins. 36 sees, by 56, checking 
by multiplication. 

10. Divide 4285 cu. yds. 6 cu. ft. 1689 cu. in. by 23, checking by 
multiplication. 

11. Also by 85, checking by multiplication. 

12. Divide 5863 gals. 3 qts. 1 pt. 3 gi. by 8, checking by multiplica- 
tion. 

13. Also by 75, checking by multiplication. 

14. How many jars, each containing 2 gals. 3 qts. 1 pt. 3 gi., can be 
filled from a cask containing 285 gals. ? Check the result by multi- 
plication. 

15. Divide 346 da. 18 hrs. 34 mins. 32 sees, by 1 da. 7 hrs. 45 mins. 
56 sees., checking the result by multiplication. 

16. A carriage wheel revolves 3 times in going 11 yds. ; how many 
times will it revolve in going of a mi. ? 

17. If 277,280 cu. in. of water weigh 10,000 Ibs., how many cubic 
feet (approximately) will weigh 1000 oz.? 

18. If a clock gains 12 mins. a day, what is the average gain per 
min. ? 

19. Supposing the distance traveled by the earth about the sun to 
be 596,440,000 mi., what is the average hourly distance traveled, 
taking the year to equal 365J da. ? 

20. Supposing the distance from the earth to the sun to be 91,713,000 
mi. and that the sun's light reaches the earth in 8 mins. 18 sees., 
what is the velocity of light per sec. ? 

21. From the data of Ex. 20 and that on p. 3, find how long it 
would take the sun's light to reach Neptune. Express the result in 
hours, minutes, and seconds. 



THE METRIC SYSTEM. 59 



II. THE METRIC SYSTEM. 

Soon after the opening of the nineteenth century, France 
legalized a uniform system of measures generally known 
as the Metric System. This is now used in 



practical business by most of the highly civi- H E 
lized nations, except the United States and J E 
England and her dependencies. In scientific 
work it is generally used by all countries, and 
there is every reason to believe that it will also 



become universal in business. a 

Units. The system is based on the unit of * 

length, called the meter (meaning measure), f 

which is 0.0000001 (or 10~ 7 ) of the distance a | 

from the equator to the pole. 5: | 

The unit of capacity is the liter, a cube 0.1 of S * 

a meter on an edge. 2. 



The unit of weight is the gram, the weight of 
cube of water 0.01 of a meter on an edge. 
Through an error in fixing the original units, they 



are not exactly as stated, but the system loses none of g 
its practical advantages on this account. The original | 
units are preserved at Paris. 

The prefixes set forth on p. 60 must be thoroughly g 
memorized, after which the metric system offers few 
difficulties. Some of these prefixes are never used with g. 
certain units in practice, just as the only units generally g. 
used in speaking of United States money are dollars and ^ 
cents. We never say, "4 eagles 2 dollars 5 dimes and - 
3 cents " for $42.53. So in the metric system the myria- 
liter and kiloliter are never used, and the dekameter and 
hektometer rarely. In the following tables, the units most commonly 
used are, therefore, printed in bold-faced type. 

The abbreviations of the metric system are not uniform even in 
France. Those here given have been adopted by the International 
Committee of Weights and Measures and by other international asso- 
ciations, and are therefore given as the most approved now in use. 



60 



HIGHER ARITHMETIC. 



THE PREFIX MEANS 

myria- 10000 
1000 
100 



TABLES. 



AS IN 



WHICH MEAXS 



kilo- 

g hekto- 
deka- 

.2 deci- 
g >3 centi- 
| milli- 

Greek. mikrO- 



10 

1 

0.1 

0.01 

0.001 



myriameter 10000 
kilogram 
hektoliter 
dekameter 

decimeter 
centigram 
millimeter 



0.000001 mikrometer 



0000 


meters. 


1000 


grams. 


100 


liters. 


10 


meters. 


1 




0.1 


of a meter. 


0.01 


of a gram. 


0.001 


of a meter. 


0.000001 


of a meter. 



TABLE OF LENGTH. 

A myriameter = 10,000 meters. 
A kilometer (km) = 1000 " 
A hektometer = 100 " 

A dekameter = 10 " 

Meter (m) 

A decimeter (dm) = 0.1 of a meter. 

A centimeter (cm) = 0.01 u 

A millimeter (mm) = 0.001 " 

A mikron 0") 0.000001 " 

TABLE OF SQUARE MEASURE. 

A square myriameter = 100,000,000 square meters. 

" kilometer (km 2 ) = 1,000,000 " 



" hektometer 

" dekameter = 

Square meter (m 2 ) 
A square decimeter (dm 2 ) = 
" centimeter (cm 2 ) = 
" millimeter (mm 2 ) = 



10,000 
100 



0.01 of a square meter. 
0.0001 
0.000001 
The square dekameter is also called an are; and since there are 

100 dm 2 in 1 hm 2 , a square hektometer is called a hektare. These 

are used in measuring land. 

TABLE OF CUBIC MEASURE. 

A cubic myriameter = 10 12 cubic meters. 

" kilometer 10 9 " 

" hektometer =1,000,000 " 

" dekameter 1000 

Cubic meter (m 3 ) 



THE METRIC SYSTEM. 61 

A cubic decimeter (dm 3 ) = 0.001 of a cubic meter. 

" centimeter (cm 3 ) = 0.000001 " 

" millimeter (mm 3 ) = 0.000000001 " 

The cubic meter is also called a stere, a unit used in measuring wood. 

TABLE OF WEIGHT. 

A metric ton (t) = 1,000,000 grams. 
A quintal (q) = 100,000 " 
A myriagram = 10,000 " 
A kilogram (kg) = 1000 " 
A hektogram 100 " 

A dekagram = 10 " 

Gram (g) 

A decigram 0. 1 of a gram. 

A centigram (eg) = 0.01 " 

A milligram (mg) = 0.001 " 

A mikrogram (7) = 0.000001 " 

The metric ton is the weight of 1 m 3 of water ; the kilogram of 
1 dm 3 or 1 liter of water ; and the gram of 1 cm 3 of water. 

TABLE OF CAPACITY. 
A hektoliter (hi) = 100 liters. 
A dekaliter =10 " 

Liter (1) 

A deciliter = 0. 1 of a liter. 

A centiliter =0.01 " 

A milliliter (ml) = 0.001 " 

A mikroliter (X) = 0.000001 " 

TABLE OF EQUIVALENTS. 

In general, the metric system is used by itself, as in scientific work, 
and the common English-American system by itself. Hence, there is 
little demand for reducing from one to the other. Such reductions 
are, however, occasionally necessary, and hence a few of the common 
equivalents are here given. These equivalents are only approximate. 

A meter = 39.37 inches = 3 feet nearly. 

A liter = 1 quart nearly. 

A kilogram = 2.2 pounds nearly. 

A kilometer = 0.62 of a mile = 0.6 of a mile nearly. 

A gram = 15.43 grains = 15| grains nearly. 

A hectare = 2.47 acres = 2| acres nearly. 



62 HIGHER ARITHMETIC. 

Oral Exercises. 1. What is the meaning of hekto- ? myria- ? 
centi-? kilo-? deci- ? deka- ? milli- ? mikro- ? 

2. What is the prefix which means 10,000? 0.1? 10? 100? 
0.01? 1000? 0.001? 

3. About when was the metric system established ? Where ? 
How extensively is it used at present, (a) in business, (6) in science ? 
What are its advantages over the older systems ? 

4. How was the length of the meter fixed ? How was the liter 
fixed ? the gram ? 

5. What is the weight of a liter of water ? (This is only approxi- 
mate and it refers to distilled water at its maximum density.) 

6. What is the weight of a cubic centimeter of water ? of a cubic 
decimeter ? of a cubic meter ? 

7. How many mm in a km ? in a hektometer ? in a myriameter ? 

8. How many cm 2 in a m 2 ? in a km 2 ? 

9. How many mm 3 in a cm 3 ? in a liter ? in a m 3 ? 

10. How many g in 125 kg ? in a metric ton ? 

11. How many dm 3 in 5 steres ? in a cubic dekameter ? 

12. Reduce 17 km to m ; to mm ; to dm ; to /*. 

13. Reduce 5 dekaliters to 1 ; to cm 3 ; to X. 

14. Reduce 300 ha to a ; to m 2 ; to km 2 . 

15. Reduce 45,000 m 2 to ha ; to a ; to a fraction of a km 2 . 

16. Reduce 0.573 m 2 to cm 2 ; to mm 2 . 

17. Reduce 15 km 2 to m 2 ; to cm 2 ; to ha. 

18. Reduce 27 m 3 to dm 3 ; to mm 3 ; to 1. 

19. 25 kg are how many Ibs. , to the nearest unit ? 

20. 300 km are how many mi. , to the nearest unit ? 

21. 65 1 are how many qts., to the nearest unit ? 

22. 30 ha are how many acres, to the nearest unit ? 

23. 50 acres are how many ha, to the nearest unit ? 

24. 20 qts. are how many 1, to the nearest unit ? 

25. 50 mi. are how many km, to the nearest unit ? 

26. 220 Ibs. are how many kg, to the nearest unit ? 

27. 325 ft. are how many m, to the nearest unit ? 

28. The Eiffel tower at Paris is 300 m high ; this is about how 
many feet ? 

29. The papers report the rainfall at Berlin, for a given period, to 
be 11.0 cm ; this is how many inches, to the nearest tenth? 

30. What is the pressure in grams per cm 2 of a column of water 
1 m deep ? 



THE METRIC SYSTEM. 63 

Written Exercises. 1. The length of a wave of sodium light 
is 5893 X 10" 8 cm j how many such wave-lengths in 1 m ? in 1 /* ? 

2. Cast copper being 8.8 times as heavy as an equal volume of 
water, what is the weight of 5 dm 3 ? 

3. A stream flowing uniformly 1 km per hr. flows how many cm 
per sec. ? 

4. A liter of mercury weighs 13.596 kg ; how many mm 3 of mer- 
cury weigh 1 g ? 

5. A man takes 120 steps in walking 100 m ; what is the average 
length of each step ? 

6. At the rate given in Ex. 5, how many steps will be taken in 
walking 6.2 km ? 

7. How many mm 2 in dm 2 ? in of a decimeter square ? 

8. Air being 0.001276 as heavy as an equal volume of water, what 
is the weight of air in a room containing 600 m 3 ? 

9. A train traveling 1 km per min. travels how many m per sec. ? 

10. Sound travels 332 m per sec. ; how long will it take it to travel 
1 km ? Answer to 0.01 sec. 

11. Granite being 2.7 times as heavy as water, what is the weight 
of a block containing 2.50 m 3 ? 

12. Show that an inch is nearly 2.54 cm, and use this equivalent 
in Exs. 13, 14. 

13. Express the following readings from a barometer in centi- 
meters : 29.9 in., 30.0 in., 30.1 in., 30.2 in. 

14. Also the following in inches, to the nearest 0.1: 71.119cm, 
73.659 cm, 74.929 cm. 

15. Olive oil being 0.92 as heavy as an equal volume of water, and 
petroleum 0.7, and alcohol 0.83, what is the weight of a liter of each ? 

16. The distance from Paris to Rouen is 136 km ; the prices of 
tickets are, 1st class 15.25 francs, 2d class 11.40 francs, 3d class 8.40 
francs ; what is the price for each class per kilometer ? 

17. The United States government lays down these regulations 
concerning foreign mail : letters weighing 15 g or less require 5 cts.; 
packets of samples of merchandise may be sent not exceeding 350 g 
in weight, 30 cm in length, 20 cm in width, and 10 cm in height ; 
express these measures in the common units. 

18. The Eiffel tower at Paris is 300 m high and cost about 
5,000,000 francs; if enough 20-franc gold pieces each 1J mm thick 
could be piled one above another to equal this sum, would the pile 
equal the height of the tower ? 



64 



HIGHER ARITHMETIC. 



BOrUNG POINT 



OF WATER 



212 



FREEZING POINT 



32 



III. MEASURES OF TEMPERATURE. 

Temperature is ordinarily measured by a thermometer 
FAHRENHEIT using one of two scales, 
(a) the Fahrenheit, used 
in this country in ordi- 
nary business, or (&)the 
Centigrade, quite gene- 
rally used in science. 

The Fahrenheit (Fah.) 
scale was suggested by 
Fahrenheit in the early 
part of the eighteenth cen- 
tury. He took for a tem- 
perature which he obtained 
by mixing ice and salt, and 
for 212 the boiling point 
FIG. i. of water, thus bringing the 

freezing point at 32, a very 
unscientific arrangement. 

The Centigrade (C., from Latin centum, hundred, and gradus, 
degree) scale was adopted by Celsius in 1742, and is often called by 
his name. It places at the freezing, and 100 at the boiling point 
of water. 

Degrees above on each scale are usually indicated by the sign + ; 
below, by the sign . 

To reduce from Fahrenheit to Centigrade. 
V 212 Fah. 32 Fah., or 180 Fah. = 100 C., 
100 C. 5 C. 

9 



r 



It must also be remem- 



bered that the freezing point of water is at -f- 32 Fah. 
E.g., + 80 Fah. corresponds to what temperature C. ? 

1. + 80 Fah. = + 80 Fah. - 32 Fah. above freezing, or 48 Fah. 
above freezing. 

KO r* 

2. lFah. =^, 

'1 C 

3. .-. 48 Fah. = 48 X - = 26.67 C. 



MEASURES OF TEMPERATURE. 65 

Also, 10 Fah. corresponds to what temperature C. ? 

1. - 10 Fah. - 32 Fah. = - 42 Fah. 

KO p 

2. lFah. = !L <p, 

3. .-. - 42 Fah. = - 42 X ^-~ = - 23.33 C. 

To reduce from Centigrade to Fahrenheit. 

-.100 C. =180 Fah. 
.-. 1 C. = 1.8 Fah. 

E.g., + 80 C. corresponds to what temperature Fah. ? 

1. 1 C. = 1.8 Fah. 

2. .-. 80 C. = 80 X 1.8 Fah. = 144 Fah. above freezing. 

3. .-. 80 C. corresponds to 144 Fah. + 32 Fah., or 176 Fah. 

Exercises. Find the temperatures, Fah. or C., corresponding 
respectively to the following temperatures, C. or Fah. 

1. 

4. 

7. 
10. 
13. 
16. 
19. 
22. 

25. Express on Centigrade scale the following melting points : 
(a) lead + 630 Fah., (6) mercury 38.99 Fah., (c) ice + 32 Fah., 
(d) silver + 873 Fah., (e) copper + 1996 Fah., (/) cast iron + 2786 
Fah., (g) tin + 455 Fah. 

26. Express on Fahrenheit scale the following boiling points : 
(a) alcohol +78 C., (6) ether +35 C., (c) mercury + 357 C., 
(d) sulphuric acid + 338 C. 

27. There is another kind of thermometer which is often used, 
known as the Reaumur thermometer, the being the freezing and 
80 being the boiling point of water. Express on the Reaumur scale 
the following melting points : (a) camphor 175 C., (6) paraffine 55 C., 
(c) phosphorus 44 C., (d) rock salt 800 C., (e) sugar crystal 170 C. 

28. Express on the Reaumur scale the following boiling points : 
(a) wood alcohol 150.8 Fah., (6) benzine 176 Fah., (c) chloroform 
141.8 Fah., (d) glycerine 554 Fah. 



+ 122 Fah. 


2. 


+ 115.8 Fah. 


3. 


- 35 Fah. 


- 30 Fah. 


5. 


+ 30 Fah. 


6. 


+ 25.7 Fah. 


+ 13.1 Fah. 


8. 


40. 7 Fah. 


9. 


+ 100. 5 Fah. 


- 1.2 Fah. 


11. 


+ 1.2 Fah. 


12. 


+ 100 Fah. 


+ 40 C. 


14. 


- 40 C. 


15. 


+ 100 C. 


- 7.8 C. 


17. 


+ 15.8 C. 


18. 


- 17.78 C. 


- 21.4 C. 


20. 


18.1 C. 


21. 


- 45 C. 


+ 45 C. 


23. 


33 C. 


24. 


- 9.6 C. 



CHAPTER VII. 
Mensuration. 



THERE are certain measurements which are so commonly 
needed in business and in science that they are generally 
considered as part of arithmetic, although the strict proofs 
of the principles involved are matters of geometry. It is, 
however, possible to arrive at the results without the use 
of demonstrative geometry, and for those who have not 
studied that science the present chapter will be of value. 

Area of a rectangle. If two sides of 
a rectangle are 3 in. and 5 in. respec- 
tively, then, in the figure, the area of the 
strip A B is 5 X 1 sq. in., and the total 
FIG. 2. area is 3 X 5 X 1 sq. in., or 15 sq. in. 

Similarly, the area of a rectangle b cm long and h cm high is 
h - b 1 cm 2 , or hb cm 2 . This is often expressed by saying that the 
area equals the product of the base and altitude, meaning that the 
abstract numbers have this relation. 

Length of a rectangle. If a rectangle has an area of 
6 cm 2 and a height of 2 cm, the number of units of 
length, say /, must be such that I 2 1 cm 2 6 cm 2 ; 

.-.l= - = 3 ;/. the length is 3 cm. 
2 1 cm 2 

Similarly, if the area is a cm 2 and the height is h cm, the number 
of units of length, say I, must be such that I h 1 cm 2 = a cm 2 ; 

.-. 1 = r - ^ = an abstract number expressing the units of length. 

h 1 cm 2 



MENSURATION. 



67 



FIG. 3. 



Area of any parallelogram. Since auy parallelogram 
has the same area as the rectangle 
of the same base and altitude, as is 
seen by cutting off the triangle T in 
the figure and placing it in the posi- 
tion T', for purposes of measurement 
of area, length, and height, a parallelogram may be con- 
sidered a rectangle. 

Area of a triangle. Since two con- 
gruent triangles, as T in the figure, 
may be so placed as to form a par- 
allelogram, therefore any triangle is 
half of a rectangle of the same base and altitude. 

Hence, the area of a triangle of base 3 cm and altitude 2 cm is 
i 2 3 1 cm 2 = 3 cm 2 . In general, if the base is b cm and the height 
is h cm, the area is - hb cm 2 . 

If the area of a triangle is 3 cm 2 and the height is 2 cm, the number 
of units of length of base, say 6, is such that 2 6 1 cm 2 = 3 cm 2 ; 

= 3 ; therefore the base is 3 cm. 




FIG. 



6 = 



2 1 cm 2 



Area of a trapezoid. If the trapezoid T in the figure 

swings about the point o to the 

position T', leaving its trace at T, 

the figure T~P is a parallelogram. 

Hence, the area of a trapezoid equals 

half the area of a rectangle of the 

same altitude and with a base equal to the sum of the two 

bases of the trapezoid. 



FIG. 5. 



Exercises. 1. Find the area of a parallelogram of which the base 
is 4 ft. 3 in. and the altitude 4 ft. 

2. Find the area of the right-angled triangle in which the two sides 
including the right angle are 5 ft. and 8 ft. 

3. Find the area of a trapezoid whose bases are 6 ft. 2 in. and 7 ft. 
4 in. and whose altitude is 4 ft. 



68 



HIGHER ARITHMETIC. 




FIG. 6. 



Abscissas and ordinates. If, in Fig. 6, XX' is perpen- 
dicular to YY' at 0, and y x to 
OX, then x x is called the abscissa 
of point P 1? and yj is called the 
ordinate of that point. 

Similarly x 2 , y 2 , are the abscissa 
and ordinate of P 2 , and so on for P 3 , 
P 4 , etc. 

To find the area of the field PiP 2 P 3 P4, the area of the trapezoid 
between yi and y 4 may be found, and from this may be subtracted 
the areas of the trapezoids between yi and y 2 , y 2 and y 3 , y 3 and y 4 ; 
this plan is sometimes used by surveyors, OY representing the north 
line and OX the east line. It is more convenient, however, to let 
OX pass through the most southern point P 3 , or OY pass through the 
most western point PI. Surveyors usually take the latter plan and 
find the areas of the trapezoids between x 3 and x 4 , x s and x 2 , etc. 

Mensuration of a square. Since a square is a particular 
kind of rectangle, the area of a square of side 3 cm is 
3-3-1 cm 2 = 9 cm 2 . And if the area is given as 9 cm 2 , 
the number of units in a side, say s, is such that s s 1 cm 2 
= 9 cm 2 , whence s 2 = 9, and s=:V9 = 3. .'. the side is 
3 cm long. 

It should again be observed that, by the ordinary definition of 
square root, the expression V9 cm 2 has no meaning. 

The Pythagorean theorem. In the figure, if the tri- 
angles 1, 2, 3, 4 are taken away, the 
square on the hypotenuse of a right- 
angled triangle remains ; and if the two 
rectangles AP, PB, are taken away from 
the whole figure, the sum of the squares 
on the two sides of the triangle remains ; 
but the four triangles together equal the 
two rectangles, hence in a right-angled 

triangle the sum of the squares on the two sides equals the 

square on the hypotenuse. 




FIG. 7. 



MENSURATION. 69 

This is known in geometry as the Pythagorean theorem because it 
is supposed to have been first proved by Pythagoras (about 500 B.C.). 
If the sides of a right-angled triangle are 3 ft. and 4 ft. , then 

1. 3 3 1 sq. ft. = 9 sq. ft., the square on one side, and 

2. 4 4 1 sq. ft. = 16 sq. ft., the square on the other side. 

3. .-. 9 sq. ft. + 16 sq. ft. = 25 sq. ft., the square on the hypotenuse. 

4. .-. the number of units in the hypotenuse = V25 = 5. 

5. .-. the hypotenuse is 5 ft. long. 

The word "equal" as here used means "have the same area as." 
For full discussion of the words "equal" and "equivalent," see 
Beman and Smith's " Plane and Solid Geometry," p. 20. 

Exercises. 1. What is the area of a floor 26 ft. 4 in. long and 
42 ft. 8 in. wide ? 

2. What is the length of a rectangular field 370 ft. wide, contain- 
ing an area of 7850 sq. rds. ? 

3. What is the length, to 0.01 m, of a square whose area is 50 m 2 ? 

4. The area of a triangle is 57 sq. in. and the height is 5 in. ; find 
the length of the base. 

5. Find, to 0.001 in., the altitude of an equilateral (equal sided) 
triangle whose side is 4 in. ; also, to 0.01 sq. in., the area. 

6. Find the length of one of the equal sides of an isosceles triangle 
whose base is 8 cm and altitude 3 cm. (An isosceles triangle is a 
triangle having two equal sides.) 

7. The area of an equilateral triangle is 10 sq. in. ; find, to 0.001 
in., the length of a side. 

8. A ship is sailing at the rate of 11.25 mi. per hr. ; a sailor climbs 
a mast 55 yds. high in 24 sees. ; find his rate of motion. 

9. A ship is sailing at the rate of 12 mi. per hr. ; a sailor walks 
across the deck at the rate of 5 mi. per hr. ; find his rate of motion. 

10. A man swims at right angles to the bank of a stream at the 
rate of 3.6 mi. per hr.; the rate of the current is 10.5 mi. per hr.; 
find the rate of the swimmer's motion. 

11. In Ex. 10, suppose the stream to be 972 ft. wide; how far 
down stream is the swimmer carried by the current ? 

12. A square and an equilateral triangle have the same perimeter ; 
how do their areas compare ? (Assume the side of the square to be 8, 
find the side of the triangle, and then find the area of each.) 

13. The hypotenuse of a right-angled triangle is 5 ft., and one side 
is 4 ft. ; show that the equilateral triangle on the hypotenuse equals 
the sum of the equilateral triangles on the two sides. 



70 HIGHER ARITHMETIC. 

14. How many square yards of plain paper are necessary to cover 
the walls and ceiling of a room 20 ft. by 15 ft. and 10 ft. high, allow- 
ance being made for three windows, each 5 ft. 4 in. by 3 ft., and one 
door 8 ft. by 3ft.? 

15. The abscissas of the several corners of a field are (in chains) 
1.00, 2.25, 6.00, 7.50, 4.30, and the corresponding ordinates are 5.10, 
0.90, 2.00, 4.70, and 7.00. Draw the figure to a scale and compute 
the area in acres. 

16. A gardener has a yard 150 ft. by 250 ft. to be laid out into 
beds ; allowing a foot all around the edge of the yard for grass, how 
many beds 4 ft. by 8 ft. can be laid out ? 

17. How many bricks 4 in. by 8 in. are needed to lay a cellar floor 
20 ft. by 36 ft. ? 

18. How many acres in a farm which consists of two rectangular 
pieces, one 95 rds. by 160 rds. and the other 75 rds. by 22 rds. ? 

19. A rectangular farm 205 rds. long and 175 rds. wide is under 
cultivation, with the exception of a piece of woods 62 rds. by 58 rds. ; 
how many acres are cultivated ? 

20. How many bunches of shingles should be bought for shingling 
a barn with a 40-ft. roof and 16-ft. rafters, laying the shingles 4 in. 
to the weather and a double row at the bottom ? The average width 
of the shingles is 4 in. and there are 250 in a bunch. 

21. How many bushels of wheat would be taken from a field 60 
rds. by 80 rds., the average yield being 30 bu. to the acre ? 

22. How much will it cost to plaster a room 25 ft. long, 20 ft. wide, 
and 12 ft. high, at 9 cts. per sq. yd., no allowance being made for 
windows, etc. ? 

23. It is shown in physics that if two forces are pulling from a 

point P and are represented in direction 
and intensity by the lines PA, PB, the 
resultant force is represented by PC, the 
diagonal of their parallelogram. Suppose 
two forces are pulling at right angles with 
the intensity 5 Ibs. and 8 Ibs., what is the 
intensity of the resulting force ? Draw 
the parallelogram to some convenient scale. 

24. As in Ex. 23, suppose the forces are each 10 kg, pulling at an 
angle of 60 (an angle of an equilateral triangle). 

25. As in Ex. 23, suppose the forces are each 3 Ibs., pulling at an 
angle of 90. 




MENSURATION. 71 

Board measure. In measuring lumber, a board 1 ft. 
long, 1 ft. wide, and 1 in. or less thick is said to have a 
measurement of 1 board foot. 

E.g., a board 16 ft. long, 12 in. wide, and 1 in. or less thick con- 
tains 16 board feet. But a board 16 ft. long, 8 in. wide, and 1-J- in. 
thick contains 16 T 8 2 f of 1 board foot, or 16 board feet. 

In speaking of lumber, the word "foot" is generally used for 
"board foot." The price of lumber is usually quoted by the 1000 
board feet ; thus, " $25 per M " means $25 per 1000 board feet. 

Exercises. 1. How many feet in each of the following boards : 
(a) 14 ft. long, 8 in. wide, 2 in. thick ? 
(6) 16 ft. long, 6 in. wide, 1 in. thick ? 

(c) 12 ft. long, 4 in. wide, 2 in. thick ? 

(d) 15 ft. long, 9 in. wide, f in. thick ? 

(e) 10 ft. long, 12 in. wide, 1 in. thick ? 
(/) 8 ft. long, 10 in. wide, 2 in. thick ? 
(g) 12 ft. long, 8 in. wide, 1 in. thick ? 
(h) 16 ft. long, 6 in. wide, 1 in. thick ? 

2. What is the cost of 20 2-in. planks, each 16 ft. long by 9 in. 
wide, @ $25 per M ? 

3. What is the cost of 50 1^-in. boards, each 12 ft. long by 6 in. 
wide, @ $30 per M ? 

4. How many feet of lumber will it take to floor a barn 30 ft. long 
and 16 ft. wide with 2-in. planks ? 

5. How many feet in a beam 18 ft. long and 8 in. square on the 
end? 

6. A 16-ft. beam costs $4 @ $30 per M ; it is square on the end ; 
what is the thickness ? 

7. How many feet in a stick of timber 16 ft. long and 10 in. 
square ? 

8. How much will it cost for 1-in. boards to fence an acre lot 8 rds. 
wide, the boards being 6 in. wide and the fence 4 boards high, the 
price of lumber being $20 per M ? 

9. How many feet of unmatched lumber will it take to cover the 
sides and gable peaks of a barn 30 ft. long and 20 ft. wide, 16 ft. high 
to the eaves, the gable peaks being 8 ft. high ? 

10. How many feet of lumber will it take to build a 6-board fence 
around a 3-acre rectangular lot 316.8 ft. wide, 6-in. boards being used 
and no allowance being made for waste in cutting ? 



72 



HIGHER ARITHMETIC. 



Circumference of a circle. By measuring the circum- 
ferences and the diameters of several circles, and taking 
the average, the circumference will be found to be about 
3| times the diameter. In geometry (see Bernan and 
Smith's " Geometry," p. 190) it is proved that the circum- 
ference is more nearly 3.14159 times the diameter. 

This number, 3.14159 , is usually represented by the 

Greek letter TT (pt). 

For practical purposes, the value of TT is usually taken 
as 3| or else as 3.1416. The latter value is used in this 
work and should be employed by the student in all compu- 
tations unless otherwise directed. 

v if c stands for the circumference, and d the diameter, and r the 
radius, c = rtd = 2 Ttr. 

.-. d = cjit. 

v c/7t is the same as -c, it is often convenient to know -, the 

7t 7t 

reciprocal of Tt. This is easily shown to be 0.3183 , and may be 

used as a multiplier instead of using it as a divisor, if desired. Carried 
to seven decimal places the value is 0.3183098. 

Area of a circle. A circle can be cut into figures 
which are nearly triangles with altitude r and with bases 
whose sum is c. Supposing the figures to be triangles, 
the area would be c r, and it is proved in geometry 
that this is the exact area. 




FIG. 8. 

v c = 2 itr, .. if a stands for the area, 

a = $cr = $-2rtr-r = ?rr 2 . 
E.g., if the radius is 3 ft., the area is 

it - 3 3 1 sq. ft. = 28.2743 sq. ft. 



MENSURATION. 



73 



Exercises. 1. Show that r = c/2 n. 

2. Show that a = xd 2 . 

3. Show that r = \la/7C, and hence that d = 2 Va/7T. 

4. Find the circumferences of circles with radii (a) 22, (6) 24.1 in., 
(c) 17.1 m, (d) 0.123 m, (e) 34 mm, (/) 10.5 ft., (g) 4 yds., each cor- 
rect to 0.001. 

5. Find the radii of circles with circumferences (a) 311.0177 cm, 
(6)207.3451 in., (c) 160.2212 m, (d) 43.9823ft., (e) 1.2566 cm, each 
correct to 0.1. 

6. Find the areas of circles with radii (a) 30 in., (6) 325 cm, (c) 
17.8 mm, (d) 425 ft., (e) 0.78 m, each correct to 0.001. 

7. Find the radii of circles with areas (a) 606,831 m 2 , (6) 636,173 sq. 
ft., (c) 7.0686cm 2 , (d) 5026.5482 sq. in., (e) 1963.4954 sq. in., each 
correct to 0.1. 

8. Find the areas of circles with circumferences (a) 163.3628 in., 
(6) 628.32 ft., (c) 889.07 cm, (d) 785.40 cm, (e) 2180.27 m, each cor- 
rect to 1. 

9. Find the circumferences of circles with areas (a) 372,845 sq. in., 
(6) 384,845 sq. ft., (c) 66,052 m 2 , (d) 61,575 cm 2 , (e) 7697.6874 sq. in., 
each correct to 0.1. 

10. Find the diameter of a wheel that makes 373 revolutions in a 
mile. (Correct to 0.1.) 

11. What should be the area of the opening of a cold-air box for a 
furnace to supply 1 hot-air pipe 1 ft. in diameter and 5 hot-air pipes 
8 in. in diameter, it being necessary that the cross area of the cold-air 
box be f that of the hot-air pipes together ? 



Volume of a rectangular parallelepiped. 

gular parallelepiped is 1 cm long, 

1 cm wide, and 1 cm high, the / / 

volume is, by definition, 1 cm 3 ; / / 

if it is 3 times as long, its volume 

is 3 1 cm 3 ; and if it is also twice 

as high, its volume is 2 3 1 cm 3 ; 

and if it is also twice as wide, 



If a rectan- 



its volume is 
12 cm 3 . 



2 -2 -3-1 cm 8 , or 



FIG. 9. 



Similarly, the volume of such a solid I cm long, w cm wide, and 
h cm high is whl cm 3 . 



74 HIGHER ARITHMETIC. 

Length of a rectangular parallelepiped. If the volume 
of the solid is 12 cm 3 , and the area is 4 cm 2 on an end, the 
number of units of length must be such that I - 4 1 cm 3 

= 12 cm 3 ; .*.= = 3} .-. the length is 3 cm. 

Similarly, if the length, 3 cm, and the volume, 12 cm 8 , are known, 
and the area of an end is required, the number of square units, a, 

12 cm 3 
must be such that 3 a - 1 cm 3 = 12 cm 3 ; .-. a = cm3 = 4 ; .-. the 

area of an end is 4 cm 2 . 

Volume of any parallelepiped. As any parallelogram 
has the same area as the rectangle of equal base and equal 
altitude, so any parallelepiped has the same volume as the 
rectangular parallelepiped of equal base and equal altitude. 




FIG. 10. 

In the figure, solid III equals solid II, and solid II equals 
solid I, a rectangular parallelepiped of equal base and equal 
altitude. 

The proof is too elaborate to be considered at this time. It is, 
however, somewhat similar to that already given as to the area 
of a parallelogram. See Beman and Smith's "Plane and Solid 
Geometry," p. 252. 



MENSURATION. 75 

Exercises. 1. What is the volume of a cube 1 in. on an edge ? 
2 in.? 4 in.? 8 in.? 

2. A cube has a volume of 6 cu. ft.; find the edge, correct to 
0.001 ft. 

. 3. A parallelepiped of altitude 6 cm has a base 2 cm wide by 3 cm 
long; what is its volume ? Similarly, if the dimensions are 4.75 cm, 
0.35 cm, 3.33 cm. 

4. On a base of 7.25 sq. ft. is a parallelepiped whose volume is 
20 cu. ft.; find the height, correct to 0.001 ft. What would be the 
height if the volume were 40 cu. ft. ? 

5. What is the cost of digging a cellar 24 ft. by 12 ft., and 3 ft. 
deep, at 40 cts. per cu. yd. ? 

6. How many cubic feet of ice can I pack in an ice house 100 ft. 
by 63 ft., 18 ft. high, allowing 3 ft. on each side and 2 ft. above and 
below for sawdust ? 

7. At 66 cts. a bushel, what is the value of the wheat which fills a 
bin 6 ft. by 5ft. by 5ft.? 

8. How many cords in a pile of 4-ft. wood 18 ft. long and 4 ft. 
high? 

9. In a set of 12 steps to a high-school building each step is com- 
posed of 4 blocks of sandstone, each block being 2-j- ft. long, 2 ft. wide, 
7 in. high ; the stone costs 60 cts. per cu. ft. , laid ; find the cost of 
the steps. 

10. How many loads (cubic yards) of earth must be taken out in 
excavating a canal 8200 ft. long, 300 ft. wide, and 16 ft. deep ? 

11. How many gallons (calling 7-J- gals, equal to 1 cu. ft.) in a tank 
2ft. by 2ft. by 1ft.? 

12. A pile of 4-ft. wood 300 ft. long and 3 ft. 4 in. high cost $125 ; 
how much did it cost per cord ? 

13. Find the surface and the volume of a cube in which the 
diagonal of each face is 15 in. 

14. The volume of a cube is 8000 cu. in. ; required the length of its 
diagonal. 

15. Find the edge of a cube whose surface equals the sum of the 
surfaces of two cubes whose edges are 120 in. and 209 in. 

16. How many tons of water will a tank 16 ft. long, 8 ft. wide, and 
7 ft. deep contain ? (1 cu. ft. of water weighs 1000 oz. nearly.) 

17. A zinc tank, open at the top, is 32 in. long, 21 in. wide, and 
16.5 in. deep, inside measure; the metal is J in. thick; required its 
weight and the weight of the water which it will hold. (Zinc is 7.2 
times as heavy as water.) 



76 



HIGHER ARITHMETIC. 




FIG. 11. 




Volume of a prism. Since any triangular 
prism equals half of a parallelepiped of 
the same altitude and of double the base, 
therefore any triangular prism equals a 
parallelepiped of the same altitude and 
equal base, and hence it equals a rectan- 
gular parallelepiped of equal base and equal 
altitude. 

And since any prism can be cut into triangular prisms, 
as in Fig. 12, therefore any prism equals a 
rectangular parallelepiped of equal base and 
equal altitude. 

Volume of a pyramid. If a hollow prism 
and a hollow pyramid of the same base and 
altitude be made from pasteboard, and the 
latter be filled three times with sand and 
the contents poured into the prism, the latter 
will be exactly full. Hence it may be inferred, as it is 
proved in geometry, that the volume 
of a pyramid is one-third the volume 
of a rectangular parallelepiped of equal 
base and equal altitude. 

E.g., the volume of a prism of alti- 
tude 2 ft. and base 3 sq. ft. is 2-3-1 
cu. ft. = 6 cu. ft. The volume of a 
pyramid of equal base and equal alti- 
tude is -j- 2 3 1 cu. ft. = 2 cu. ft. 

Volumes of a cylinder and a cone. If a hollow rectan- 
gular parallelepiped, a hollow cylinder, and a hollow cone 
be constructed from pasteboard, all having equal bases and 
equal altitudes, then, by the method employed for finding 
the volume of a pyramid, the cylinder will be found to have 
the same volume as the rectangular parallelepiped, and the 
cone to have one-third of that volume. 




MENSURATION. 



77 



In our work only cones and cylinders, whose bases are circles, 
will be considered. In the case of right circular cones and cylinders, 
the curved surfaces are easily measured. For the surface of each 
may be imagined as unrolled from the solid itself, in which case 
the surface of the cylinder will unroll into a rectangle, and that of 
the cone will unroll into a sector of a circle. As a circle has the 
same area as a rectangle whose base equals the circumference of the 
circle and whose altitude equals half the radius, so the curved surface 
of the cone equals a rectangle whose base equals the circumference of 
the base of the cone and whose altitude equals half the slant height 
of the cone. Students are advised to construct the figures from paste- 
board and to read this paragraph with the solids in hand. 

Surface of a sphere. If the surface of a hemisphere be 
wound by a waxed tape, 
as in the figure, it will 
be found to take twice as 
much tape as is needed 
to wind a great circle of 
the sphere. Therefore, 
the area of the surface FIG. 14. 

equals that of four great circles, or 4 Trr 2 . 

E.g., the surface of a sphere whose radius is 5 ft. is 
4 TT 5 2 - 1 sq. ft. = 314.16 sq. ft. 

Volume of a sphere. If three points on a sphere be 
connected with the center, a solid is cut 
out which is nearly a pyramid with altitude 
equal to the radius r. If the sphere be 
divided into any number of such solids, 
the sum of the bases is the surface of the 
sphere, 4 Trr 2 , and the common altitude is r. 
Therefore the volume of all these solids, 
considered as pyramids, is -J r 4 Trr 2 , or f Trr 3 . In geom- 
etry it is strictly proved that this is the volume. 

E.g., the volume of a sphere whose radius is 5 ft. is 
| IT 5 1 cu. ft. = 523.6 cu. ft. 





FIG. 15. 



78 HIGHER ARITHMETIC. 

Exercises. 1. Find the volume of a pyramid whose base is an 
equilateral triangle 3 in. on an edge, and whose altitude is 5 in., 
correct to 0.01 cu. in. 

2. What is the volume of the pyramid of Cheops, its base being 
764 ft. square and its altitude being 480.75 ft. ? 

3. What is the weight of a wall 5.40 m long, 2.30 m high, and 
0.50 m thick, the wall being entirely composed of stone which is 1.83 
times as heavy as water. 

4. What is the volume of a cylinder of radius 2 in. and altitude 
5 in.? 

5. What is the volume of a cone of radius 8 in. and altitude 
9 in.? 

6. The circumference of a steel shaft is 5 dm ; what is the length 
of its radius ? 

7. If the shaft in Ex. 6 is 8 m long, what is its volume ? What is 
its weight, steel being 7.8 times as heavy as water ? 

8. What is the area of the entire surface of a cone whose radius is 
5 in. and whose slant height is 4 in. ? 

9. Find the surfaces of the spheres of radii (a) 3 in., (6) 15 dm, 
(c) 51 cm, (d) 201 mm, (e) 415 mm, each correct to 6 significant 
figures. 

10. Find the volumes of the spheres of radii (a) 1 ft., (6) 6 in., 
(c) 42 cm, (d) 66 mm, (e) 123 mm. 

11. Find the radii of the spheres of surfaces (a) 3.1416 sq. in., 
(6) 706.8583 sq. in., (c) 5026.5482 m 2 , (d) 7853.9816 sq. in., (e) 70,686 
mm 2 , each correct to 0.1. 

12. Find the radii of the spheres of volume (a) 1,436,750 cm 8 , 
(6) 523,600 cu. in., (c) 195,432 m 3 , (d) 4,188,800 mm 3 , (e) 7,979,250 
mm 3 , correct to units' place. 

13. If s = the surface, and c = the circumference, and v = the 
volume of a sphere of radius r, show that r = Vs/^ s = c 2 /;r, 
c VTTS, and r = V3 v/l it. 

14. Supposing 1 cu. ft. of water to weigh 1000 oz. and marble to 
be 2.7 times as heavy as water, find the weight of a sphere of marble 
3 ft. in circumference, correct to 0.1 oz. 

15. A cylindrical cistern 2 m in diameter is filled with water to 
the depth of 2 m ; what is the weight of the water ? 

16. A cylindrical tank is 1.20 m in diameter and 3 m long ; it is 
filled with petroleum, which is 0.7 as heavy as water j what is the 
weight of the petroleum ? 



MENSURATION. 79 

17. The diameter of a sphere and the altitude of a cone are equal 
and they have equal radii ; how do their volumes compare ? 

18. Find the radius of that circle the number of square inches 
of whose area equals the number of linear inches of its circum- 
ference. 

19. Find the radius of that sphere the number of square inches of 
whose surface equals the number of cubic inches of its volume. 

20. The radius of the earth being approximately 4000 mi., what is 
its approximate area, correct to 1000 sq. mi.; also its approximate 
volume, correct to 1000 cu. mi. 

21. From Exs. 14 and 20, compute the weight of the earth to 
three significant figures, expressing the result in the index notation, 
knowing that the earth is 5.6 times as heavy as water. 

22. What is the weight of a column of water in a pipe 10 m high, 
of interior diameter 0.09 m ? 

23. The surface of a pyramid is made up of equilateral triangles 
3 in. on a side ; find the area. 

24. In Ex. 23, given that the perpendicular from the vertex of the 
pyramid meets the base two-thirds of the way from any of its vertices 
to the opposite side, find the volume. 

25. What is the length of the diagonal of a cube whose edge is 
10 in. ? 

26. What is the length of the edge of a cube whose diagonal is 
10 in. ? 

27. A cube is inscribed in, and another cube is circumscribed 
about, a sphere whose diameter is 10 in.; find (a) their respective 
volumes, (&) the areas of their respective surfaces. 

28. What is the cost of 100 km of copper wire 0.53 cm in diameter, 
at 25 cts. a kilogram, the wire being 8.8 as heavy as water ? 

29. It is proved in geometry that the volume of a frustum of a 
pyramid or cone (that portion cut off by a plane parallel to the base) 
is equal to | h (bi + 6 2 + V&i& 2 ) cubic units, where h = the number of 
units of height, and 61, 6 2 the number of square units in the two bases. 
Draw the figure and find the volume of a frustum of a pyramid whose 
bases are squares 6 in. and 8 in. on a side, the altitude being 6 in. 

30. Also of a frustum of a cone the radii of whose bases are 3 in. 
and 5 in. , the altitude being 6 in. 

31. Also of a frustum of a pyramid whose bases are regular hexa- 
gons with sides 4 in. and 6 in., respectively, the altitude being 5 in. 

32. From a cube of wood the largest possible sphere is turned ; 
what portion of the cube has been cut away ? Answer to 0.001. 



CHAPTER VIII. 
Longitude and Time. 



THE subject of longitude and time is of practical value 
to two classes of people, navigators and astronomers. 
While, therefore, it is strictly a part of astronomy or of 
mathematical geography, custom has assigned to the ele- 
ments of the subject a place in arithmetic. Since the por- 
tions relating to Standard Time and the so-called Date 
Line belong to the general store of information required 
by every student, and since the theory forms an interesting 
application of arithmetic, an entire chapter is assigned to 
the subject. 

The prime meridian, that is, the meridian from which 
longitude is reckoned, is generally taken as the one passing 
through Greenwich, England. 

In 1675 the Eoyal observatory of England was erected at Greenwich, 
a city just below London on the Thames. As the shipping interests 
of England increased and gradually surpassed those of other nations, 
the British government was called upon to prepare large numbers of 
maps, and in time found it more convenient to use the meridian pass- 
ing through the Royal observatory than the one through the Canary 
Islands which the ancients had used. This was a signal for several 
other nations which had continued to use the old prime meridian to 
use the respective ones passing through their own observatories. In 
1884 an International Meridian Congress was held at Washington, 
and since this meeting most nations, excepting France, have used the 
Greenwich meridian. 



LONGITUDE AND TIME. 81 

From the prime meridian longitude is reckoned east and west to 
180. West longitude is designated by the symbol + or by the letter 
W. ; east longitude by or E. If the longitudes of two places are 
+ 45 and + 100, the difference is + 100 - (+ 45) = 55 ; if the 
longitudes are 10 and 50, the difference is 10 (50) = 40; 
if the longitudes are + 45 and 50, the difference is + 45 (50) 
= 95. In other words, if both longitudes are east, or if both are 
west, subtract ; if one is east and the other west, add. 

The two tables of longitude and time. Since the earth 
revolves upon its axis once in 24 hours, the place in which 
we live passes through 360 in that length of time. Hence 
the following tables : 





TABLE I. 






360 correspond to 


24 hrs. 






1 corresponds to 3 i 


\-y of 24 hrs., or ^ 


[ 7 hr., 


or 4 mins. 


1' corresponds to - 


^ of 4 mins. , or ^ 


L 7 min. , or 4 sees. 


1" corresponds to g 1 


L 7 of 4 sees., or -j 


^ sec. 





TABLE II. 

v 24 hrs. correspond to 360. 

.-. 1 hr. corresponds to J? f 360 or 15 - 

.. 1 min. corresponds to -^ of 15, or , or 15'. 

.-. 1 sec. corresponds to ^ of 15', or ', or 15". 

The use of these tables may be illustrated by two 
examples. 

Problem. The difference in longitude between two places is 75 
10' 30" ; what is the difference in local time ? 
Solution. 1. 75 T ^ hr. = 5 hrs. 

2. 10-4 sees. = 40 sees. 

3. 30 T ^ sec. = 2 sees. 

4. .-. the difference in time is 5 hrs. 42 sees. 
Analysis, v 1 corr. to T ! 5 hr., 75 corr. to 75 -^ hr., or 5 hrs. 

Similarly for steps 2, 3. Step 1 might also be stated, 75 4 mins. 
= 300 mins. = 5 hrs. ; and step 2 might be stated, 10 ^ min. = f- min. 
= 40 sees. The student should choose the more advantageous multi- 
plicand. 



82 



HIGHER ARITHMETIC. 



Problem. The difference in local time between two places is 
6 hrs. 12 mins. 2 sees. ; what is the difference in longitude ? 



Solution. 



6 



Analysis. 



15 = 90. 
12-i= 3. 
2 -15" =30". 

.-. the difference in longitude is 93 30". 
1 hr. corr. to 15, 6 hrs. corr. to 6-15, or 90. Simi- 
larly for steps 2, 3. What other solutions could have been used in 
steps 2, 3 ? 

Checks. The method of checking either type of problem is evidently 
found in solving the inverse problem. 

In scientific works the longitude of a place is frequently 
indicated by quoting the difference in time between that 
place and Greenwich. 

Thus, the longitude of Brussels is given as either 4 22' 9" or 
17 mins. 28.6 sees. 

TABLE OF LONGITUDES FOR REFERENCE IN SOLVING THE 
SUBSEQUENT PROBLEMS. 



Albany, N. Y. + 73 44' 48" 
Ann Arbor, Mich. + 83 43' 48" 
Athens, Greece - 23 43' 55.5" 
Berlin, Germany 13 23' 43.5" 
Berne, Switz. 7 26' 30" 
Bologna, Italy 11 21' 9" 
Brussels, Belgium 4 22' 9" 
Cairo, Egypt - 31 17' 13.5" 

Cambridge, Eng. 5' 40.5" 
Cambridge, Mass. +71 T 45" 
Cape Town, Africa - 18 28' 45" 
Chicago, 111. + 87 36' 42" 

Dublin, Ireland + 6 20' 30" 



Honolulu, Hawaii + 157 51' 48'' 

Jerusalem 35 13' 25' 

Lisbon, Portugal 

Madras, India 

Madrid, Spain 

Melbourne 

New York 

Paris, France 

Peking, China 



San Francisco 



+ 9 11' 10.5" 

- 80 14' 51" 
+ 3 41' 21" 
144 58' 42" 
+ 73 58' 25.5" 

- 2 20' 15" 
-116 27' 0" 
+ 122 25' 40.8" 



Sydney, Australia - 151 12' 39" 
Tokyo, Japan 139 42' 30" 
W'mst'n, Austral. 144 54' 42" 



Exercise. What is the difference between the longitudes and 
between the local times of : (a) Berne and Chicago ? (6) Dublin and 
San Francisco ? (c) Melbourne and Madrid ? (d) New York and 
Cambridge, Eng. ? (e) Tokyo and Cape Town ? (/) Ann Arbor and 
Berlin ? (g) Bologna and Melbourne ? (h) Brussels and Jerusalem ? 
(i) Cairo and Williamstown ? (j) Lisbon and Madras ? 



LONGITUDE AND TIME. 



83 



Sept 21 Sept. 20 
1 A M Midnight-*-]! p.M. 
-105i^^J^^ 10P.M. 

" r6 9P.M. 
45 



8 P.M. 



-30 



7 P.M. 




4-1 20 C 

10A.M. +105 3 +9Q c 
11 A.M. _ N oon 
-^ ^- g. Sept. 20 ^ ^ 

S(J AT 4p* rH T URNS TO THE **^ ftj i 

The above figure represents the earth when it is noon 
Sept. 20, on the meridian + 90, as seen from a point above 
the north pole. It is, therefore, midnight directly opposite, 
that is, at 90, and it is A.M. to the left of the noon line 
and P.M. to the right. The student should now consider 
the following questions : 

1. In the figure, how do we know that it is P.M. in New York ? 

2. Also A.M. in California, Alaska, Hawaii, Japan ? 

3. If it is 11 P.M. Sept. 20 at 75, what date is it on the other 
side of midnight, at 105 ? 

4. Hence, what date is it in Japan ? 

5. What date is it in California ? 

6. Hence, there must be a line in the Pacific ocean which separates 
what two dates ? 



84 HIGHER ARITHMETIC. 

The answers to the questions on p. 83 show the necessity 
for a line at which Sept. 21, and in general each new day, 
shall begin on the earth. This line is, by common consent, 
usually taken as nearly coinciding with the 180 meridian, 
ships changing their calendars one day on crossing this 
line. 

Exercises. 1. Draw a map similar to that on p. 83, but without 
details (a) showing the arrangement of days and hours when it is 
noon Nov. 25 at 15 ; (b) showing the same for 90 ; (c) also for 
0; (d) also for 180; (e) also for -150; (/) also for +150. 
(Separate map for each case.) 

2. When it is noon, local time, Jan. 1, 1900, at San Francisco, 
what is the date and the local time at (a) New York ? (6) Greenwich ? 

(c) Cape Town ? (d) Melbourne ? Draw a map, without details, illus- 
trating the problem. 

3. When it is Sunday noon, local time, at Sydney, what is the day 
and the local time at (a) Cairo ? (6) Dublin ? (c) Chicago ? (d) Hono- 
lulu ? (e) Peking ? (/) Greenwich ? Draw a map, without details, 
illustrating the problem. 

4. What is the longitude of that place at which, when it is noon at 
Greenwich, it is (a) 6 P.M.? (6) 2 A.M.? (c) 5 o'clock 20 min. A.M.? 

(d) 7 o'clock 50 min. 10 sec. P.M.? 

5. What is the difference between the time of Greenwich and the 
local time of: (a) Albany? (6) Athens? (c) Cambridge, Eng. ? 
(d) Honolulu ? (e) Melbourne ? (/) Madras ? (g) Jerusalem ? (h) Chi- 
cago ? 

6. What is the difference between the longitudes and between the 
local times of: (a) Williamstown and Honolulu? (6) Peking and 
San Francisco? (c) New York and Sydney? (d) Melbourne and 
Chicago ? 

7. When it is noon local time at Chicago, on what meridian is it 
midnight ? 

8. A ship at sea finds that it is 3 hrs. 34 mins. P.M. by its Green- 
wich chronometer when the sun is on the meridian ; what is the ship's 
longitude ? 

It is suggested that the teacher ascertain the approximate longitude 
of the place in which he is teaching and add this to the table. The 
number of problems which can be made from the table is limitless, 
the above being merely types. 



LONGITUDE AND TIME. 



85 



Standard time. In 1883 the railways of the United 
States and Canada proposed a system of uniform time 
which has since been quite generally adopted, not only in 
this country, but throughout the whole civilized world. In 
the United States and Canada that section of country 
lying about 7-J east and west of + 75 uses the time of 
+ 75 ; that section lying about 7-J- on either side of + 90 
uses the time of +90; and similarly for the meridians of 
+ 105, -fl20 , and +60. Since the movement was 
primarily for the accommodation of the railways, the lines 
of division are not exactly 7-J- on either side of the hour 
meridians, but are usually passed through the leading rail- 
way termini in the vicinity. 

PACIFIC TIME +120 MOUNTAIN TIME 4105 CENTRAL TIME +90 EASTERN TIME -4-75 




Exercises. (The following exercises refer to standard time only.) 

1. When it is noon at New York, what is the time at Chicago, San 
Francisco, Denver, New Orleans, Boston ? 

2. When it is 11 hrs. 30 mins. P.M. at San Francisco, what is the 
time at Denver, Milwaukee, Detroit, Albany, Philadelphia ? 

3. When it is 1 o'clock 15 mins. A.M. at Boston, what is the time 
at New Haven, Cincinnati, St. Louis, Portland, Oregon, Colorado 
Springs ? 



86 HIGHER ARITHMETIC. 

At present, the following are some of the countries using standard 
time based on the hour meridians (multiples of 15) from Greenwich, 
but the movement is so recent that the list is not intended to be 
complete. 

0, or west European time : Great Britain, Holland, Belgium. 

15, or mid-European time : Norway, Sweden, Denmark, Ger- 
many, Austria, Switzerland, Italy, Servia, Western Turkish railways. 

30, or east European time : Bulgaria, Roumania, railways to 
Constantinople. Also Natal. 

120 : western Australia. 

135 : southern Australia, Japan. 

150 : eastern Australia. 

France and Algiers use uniform time, that of Paris. Cape Colony 
(South Africa) uses 22|. 

The following exercises refer to standard time, except as 
otherwise stated. 

Exercises. 1. When it is 9 A.M. Nov. 20 at Chicago, what is the 
date and time at London, Philadelphia, San Francisco, New Orleans, 
Montreal, Denver, Melbourne ( 150 time), Venice, Constantinople 
( 30 time) ? 

2. When it is 11 P.M. Dec. 31, 1899, at New York, what is the date 
and time at Denver, San Francisco, St. Louis, Boston, London, Berlin, 
Tokyo ? 

3. When it is Sunday noon at San Francisco, what is the day and 
time at Tokyo, Williamstown ( 150 time), Rome, Amsterdam, New 
York? 

4. When would a press telegram sent from Berlin at 2 A.M. Jan. 1 
reach Portland, Oregon, if transmitted without any delay ? When 
would one sent from Melbourne at 2 A.M. Jan. 1 reach San Francisco, 
if transmitted without delay ? 

5. What is the difference between the local and standard time at 
each of the following places : (a) New York ? (6) Chicago ? (c) Cape 
Town? (d) Cambridge, Mass.? (e) Brussels? (/) Berne? (g) Berlin? 

6. At what hour (standard time) does your arithmetic class meet ? 
What is, then, the time at Glasgow, The Hague, Stockholm, Lucerne, 
Naples, Melbourne, Yokohama, Cape Town ? 

7. Twenty-four hour clocks (by which 1 P.M. is 13 o'clock, etc.) are 
used in certain parts of the world ; when it is 18 o'clock in Italy, 
what time is it in Belgium ? at Chicago ? 



CHAPTEE IX. 
Ratio and Proportion. 



I. EATIO. 

THE ratio of one number, a, to another number, b, of the 
same kind, is the quotient - 

Thus, the ratio of $2 to $5 is |r or f, or 0.4, 

$5 

and the ratio of 4 ft. to 2 ft. is ^^ or f , or 2. 

i it. 

But there is no ratio of 4 m to 3, or $5 to 2 ft., or 2 to $10. 

A ratio may be expressed by any symbol of division, 
e.g., by the fractional form, by -f-, by /, or by : ; but the 
symbols generally used are the fraction and the colon, 

as - or a : b. 
o 

The ratio - is called the inverse of the ratio 7 
a o 

If two variable quantities, a, b, have a constant ratio r, 
one is said to vary as the other. 

E.g., the ratio of any circumference to its diameter is it\ hence, a 
circumference is said to vary as its diameter. 

If - = r, then a = rb. The expression " a varies as 6" is some- 
times written a <x 6, meaning that a = rb. 
If a r -i a is said to vary inversely as b. 



88 HIGHER ARITHMETIC. 

If two variable quantities, a, b, have the same ratio as 
two other variable quantities, a', b', then a and b are said 
to vary as a' and b'. And if any two values of one variable 
quantity have the same ratio as the corresponding values 
of another variable quantity which depends on the first, 
then one of these quantities is said to vary as the other. 

E.g., the circumference c and diameter d of one circle have the 
same ratio as the circumference c' and diameter d' of any other circle ; 
hence, c and d are said to vary as c' and d'. 

If two rectangles have the same altitude, their areas depend on 
their bases ; and since any two values of their bases have the same 
ratio as the corresponding values of their areas, their areas are said to 
vary as their bases. 

The theory of ratio has its applications in geometry, in 
physics, and in practical business. 

Applications in geometry. Similar figures may be 
described as figures having the same shape, such as lines, 
squares, triangles whose angles are respectively equal, 
circles, cubes, or spheres. It is proved in geometry that 
in two similar figures 

(1) Any two corresponding lines vary as any other two 
corresponding lines. 

(2) Corresponding areas vary as the squares of any two 
corresponding lines. 

(3) Corresponding volumes vary as the cubes of any two 
corresponding lines. 

E.g., in the case of two spheres, the circumferences vary as the 
radii, the surfaces vary as the squares of the radii, the volumes vary 
as the cubes of the radii. 

These facts are easily proved. Let s, s' stand for the surfaces of 
two spheres of radii r, r', respectively. Then, 
s = 4 nrr 2 , and s' = 

s _ 4 Ttr 2 _ r 2 
* ~ ~ r' 2 



Hence, the surfaces vary as the squares of the radii. In like manner 
the volumes might be considered. 



RATIO. 89 

Exercises. 1. The ratio of 2 to x is 5 ; find x. 

2. Find x in the following ratios : (a) ^ = 7, (6) 4 : x = 9, (c) x : 17 
= 10, (d) f = M) * = Tfe:* 

3. Find x in the following ratios: (a) ^ = 2.4, (6) 7: = 4. 9, 
(c) z:5 = f, (d) | = f, (e) | = f 

4. The surfaces of a certain sphere and a certain cube have the 
same area ; find, to 0.01, the ratio of their volumes. 

5. In drawing a circle of radius 100 ft. on a scale of T ^ what 
length would represent the side of a square inscribed in the circle ? 

6. If the distance between two cities 256 mi. apart appears on a 
map as 2.56 in., what is the scale on which the map is drawn ? 

7. One cube is 1.2 times as high as another; find the ratio of 
(a) their surfaces, (6) their volumes. 

8. At that time of the day when the length of a man's shadow is 
of his height, the shadow of a telegraph pole was found to be 27.8 ft. ; 
find the height of the pole. 

9. If a sphere of lead weighs 4 Ibs., find the weight of a sphere 
of lead of (a) twice the volume, (6) twice the surface, (c) twice the 
radius. 

10. Explain Newton's definition of number : Number is the abstract 
ratio of one quantity to another of the same kind. What kinds of 
numbers are represented in the following cases : 5 ft. : 1 ft. , 1 ft. : 5 ft. , 
the diagonal to the side of a square, the circumference to the diameter 
of a circle ? 

11. Two lines are respectively 7.9 m and 23.7 m long; what is the 
ratio of the first to the second ? the second to the first ? 

12. Two arcs of the same circumference are respectively 85 31' 22" 
and 30 IT 27.7"; express the ratio of the first to the second, correct 
to 0.001. 

13. The equatorial radius of the earth is 6,377,398 m, and the 
polar radius 6,356,080 ; find the ratio of their difference to the former, 
correct to 0.01. 

14. The depths of three artesian wells are as follows : A 220 m, 
B 395 m, C 543 m ; the temperatures of the water from these depths 
are : A 19.75 C., B 25.33 C., C 30.50 C. From these observations 
is it correct to say that the increase of temperature is proportional 
to the increase of depth ? If not, what should be the temperature at 
C to have this law hold ? 



90 HIGHER ARITHMETIC. 

Applications in business. Of the numerous applications 
of ratio in business, only a few can be mentioned, and not 
all of these commonly make use of the word " ratio.' 7 

In computing interest, the simple interest varies as the 
time, if the rate is constant ; as the rate, if the time is 
constant ; as the product of the rate and the number repre- 
senting the time in years (if the rate is by the year), if 
neither is constant. 

I.e., for twice the rate, the interest is twice as much, if the time is 
constant ; for twice the time, the interest is twice as much, if the rate 
is constant; but for twice the time and 1.5 times the rate, the inter- 
est is 2- 1.5 as much. 

The common expressions " 2 out of 3," " 9 out of 10," 
" 2- to 5," " 6 per cent " (merely 6 out of 100) are only 
other methods of stating the following ratios of a part to 
a whole, f , T 9 o, f , T fa, or the following ratios of the two 
parts, f , f , f , sV 

E.g., to divide $100 so that A shall receive $2 out of every $3 is to 
divide it into two parts 

(a) having the ratio 2 : 1, or 

(6) so that A's share shall have to the whole the ratio 2 : 3, or 

(c) so that B's share shall have to the whole the ratio 1 : 3. 

To divide $1000 in the ratio of 7 : 8. 

1. There are 15 parts of which 7 are to go into one share. 

2. .-. = T 7 7 ; and v the dividend x is the product of the 

!jp J.UUU 
divisor $1000 and the quotient T 7 ^, 

3. ... x = T V of $1000, or $466.67. 

4. .-. the other share is $1000 - $466.67 - $533.33. 

Other applications will be seen in the following exercises. 

Exercises. 1. Divide $1000 so that A shall have $7 out of 
every $8. 

2. Divide $500 between A and B so that A shall have $0.25 as 
often as B has $1.25. 

3. The area of the United States is 3,501,000 sq. mi., and the 
area of Russia is 8,644,100 sq. mi.; express the ratio of the former tq 
the latter as a fraction with the denominator 100. 



RATIO. 



91 



4. From the following table of German statistics find, to 0.01, the 
ratios of the numbers of crimes for each pair of years, and also of the 
prices of grain for the same periods. 



DATE. 

1882 
1885 
1888 
1891 



CHIMES. PKICE OF GRAIN. 

535 152.3 marks per kg. 

486 140.6 " 

459 134.5 " 

511 211.2 " 



5. The following table gives statistics of workers in the woolen 
trade in 1895. Find the ratios of the expenditures for each of the 
three purposes, to the incomes, in each country, correct to 0.001. 





INCOME. 


EXPENDITURES. 


TOBACCO. 


INTOXICANTS. 


RELIGION. 


United States 


$663.13 


$9.36 


$18.39 


$8.37 


Great Britain 


515.64 


9.07 


16.01 


6.34 


France 


424.51 


7.01 


33.72 


3.25 


Germany 


275.99 


3.08 


11.74 


1.19 



6. The white population of the United States in 1780 was 2,383,000 ; 
in 1790, 3,177,257 ; in 1880, 43,402,970 ; in 1890, 54,983,890. What 
is the ratio of the population in 1790 to that in 1780 ? in 1890 to that 
in 1880 ? (Each correct to 0.01.) 

7. Before the city of Munich had sewers and an abundant supply 
of pure water, the annual death rate from typhoid fever was 242 out 
of 100,000; after these improvements the rate sank to 14 out of 
100,000. If the death rate from this disease in a town without these 
improvements is 18 out of 10,000, what would it be with the improve- 
ments and with the same rate of decrease as in Munich ? 

8. The number of women employed in the United States was 

1870. 1890. 1870. 1890. 

in art 412 10,810 in music 5,753 34,519 

as authors 159 2,725 as stenographers 7 21,185 

as journalists 35 88 as teachers 84,047 245,965 

Find, correct to 0.1, the ratios of those employed in the several 

branches in 1890 to those in 1870. 

9. In 1860, out of the $316,242,423 of our exports, $40,345,892 
represented manufactured products; in 1890, out of $845,293,828, 
$151,102,376 represented manufactured products. Find, correct to 
0.01, the ratio of the manufactured to the total in each year. 



92 HIGHER ARITHMETIC. 

Applications in physics, (a) Specific gravity. The 
specific gravity of any substance is the ratio of the weight 
of that substance to the weight of an equal volume of some 
other substance taken as a standard. 

Iii the case of solids and liquids, distilled water is usually taken as 
the standard. Thus, the specific gravity of mercury, of which 1 1 
weighs 13.596 kg, is 13.596, because this is the ratio of the weight of 
a liter of the substance to the weight of a liter of water ; 
i.e., 13.596 kg : 1 kg = 13.596. 

In the case of gases, either hydrogen or air is usually taken as the 
standard. 

The following table will be needed for reference in solv- 
ing the exercises. 

SPECIFIC GRAVITIES, REFERRED TO WATER. 

Copper 8.9. Nickel 8.9. Cork 0.24. Alcohol 0.79. 
Gold 19.3. Silver 10.5. Granite 2.7. Petroleum 0.7. 
Lead 11.3. Sulphur 2.0. Steel 7.8. Mercury 13.596. 

SPECIFIC GRAVITIES, REFERRED TO HYDROGEN. 
Air 14.43. Oxygen 15.95. Coal gas 6. 

SPECIFIC GRAVITIES, REFERRED TO AIR. 
Oxygen 1.11. Hydrogen 0.07. Chlorine gas 2.44. 

WEIGHTS OF CERTAIN SUBSTANCES. 
1 1 of water, 1 kg. 11 of air, 1.293 g. 

1 cm 3 of water, 1 g. 1 cu. ft. of water, about 62.5 Ibs., 

or about 1000 oz. 

Example. What is the weight of 1 cu. in. of copper ? 

1. 1 cu. ft. of water weighs 1000 oz. 

1000 oz. 

2. .-. 1 cu. in. of water weighs .,,.-- 

LiZo 

, 8.9- 1000 oz. 

3. .-. 1 cu. in. of copper weighs -- = 5.15 oz. 



Exercises. 1. What is the weight of a cubic foot of copper ? of 
gold? of lead? 

2. What is the weight of 1 cm 8 of nickel ? of silver ? of sulphur ? 

3. What is the weight of 1 dm 3 of cork ? of granite ? of steel ? 



RATIO. 93 

4. What is the weight of 1 1 of alcohol ? of petroleum ? of mer- 
cury ? 

5. How could the theory of specific gravity be applied to finding 
the purity of milk, the average specific gravity of good milk being 
1.032 ? 

6. What is the specific gravity of air, referred to air ? referred to 
hydrogen ? referred to water ? 

7. What is the specific gravity of coal gas, referred to air ? 

8. Sodium has a specific gravity of 1.23, referred to alcohol ; what 
is its specific gravity, referred to water ? 

9. What is the weight of 1 dm 3 of oxygen ? of hydrogen ? 

10. A cubic foot of green oak weighs 73 Ibs., of iron 432 Ibs.; find 
the specific gravity of each, correct to 0.01. 

11. An empty balloon weighs 1200 Ibs.; if 1 cu. ft. of air weighs 
1.25 oz., how many cubic feet of gas, of specific gravity 0.52 with 
respect to air, must be introduced before the balloon will begin to 
ascend ? 

12. The specific gravity of sea-water is 1.026 ; how many cubic 
feet weigh one ton ? 

13. A flask holds 27 oz. of water; what is the weight of alcohol 
that it will hold ? of petroleum ? of mercury ? 

14. In a liter jar are placed 1 kg of lead and 1 kg of copper ; what 
volume of water is necessary to fill the jar ? 

15. A nugget of gold mixed with quartz weighs 0.5 kg ; the specific 
gravity of the nugget is 6.5, and of quartz 2.15 ; how many grams of 
gold in the nugget ? 

16. A vessel containing 1 1 and weighing 0.5 kg is filled with 
mercury and water ; it then weighs, with its contents, 3 kg ; how 
many cm 3 of each in the vessel ? 

17. The specific gravity of a certain liquid is 2.000 at 0, 1.950 at 
10, 1.300 at 100; find the volume of 100 g of the* liquid at each of 
these temperatures. 

18. A cylindrical vessel 1 dm in diameter is filled with mercury 
to the height of 8 cm ; what is the pressure, in grams, upon the base ? 

19. What must be the height of a column of mercury to exert a 
pressure of 0. 5 kg per cm 2 ? 

20. The specific gravity of ice is 0.92, of sea- water 1.025 ; to what 
depth will a cubic foot of ice sink in sea-water ? 

21. From Ex. 20, how much of an iceberg 500 ft. high would show 
above water, the cross section being supposed to have a constant 
area? 



94 HIGHER ARITHMETIC. 

(b} Law of levers. If a bar AB rests on a fulcrum F 
p' w , and has a weight w at B ? 

then by exerting enough 
Y pressure p at A the weight 
can be raised. In the first 
figure the pressure is down- 



p > ward (positive pressure) ; 

A ' | A in the second it is upward 

l v (negative pressure). 

There is a law in physics 

that, if p', w' represent the number of units of distance AF, 
FB, respectively, and p, w the number of units of pressure 

and weight, respectively, then *-*-, = 1. 

In the first figure p, w, p', w' are all considered as positive ; in the 
second figure p is considered as negative because the pressure is 
upward, and w' is considered as negative because it extends the other 
way from F. Hence, the ratio pp' : ww' = 1 in both cases. 

Example. Suppose AF = 25 in., FB = 14 in., in the first figure ; 
what pressure must be applied at A to raise a weight of 30 Ibs. at B ? 

1. By the law of levers ^ = 1. 

2. .-. p = 16.8, and .-. the pressure must be 16.8 Ibs. 

Exercises. 1. Two bodies weighing 20 Ibs. and 4 Ibs. balance at 
the ends of a lever 2 ft. ong ; find the position of the fulcrum. 

2. The radii of a wheel and axle are respectively 4 ft. and 6 in. ; 
what force will just raise a mass of 56 Ibs., friction not considered ? 

3. What pressure must be exerted at the edge of a door to counter- 
act an opposite pressure of 100 Ibs. halfway from the hinge to the 
edge ? one-third of the way from the hinge to the edge ? 

4. The length of the spoke of a capstan is 6 ft. measured from the 
axis, and the radius of the drum is 1 ft. ; find the weight of an anchor 
that can just be raised by 6 men, each exerting a force equal to 100 Ibs. 
at the end of a spoke, friction not considered. 

5. In each figure, what must be the distance AF in order that a 
pressure of 1 kg may raise a weight of 100 kg 3 dm from F ? 



RATIO. 95 

(c) Boyle's law. It is proved in physics that if p is the 
number of units of pressure of a given quantity of gas, and 
v is the number of units of volume, then p varies inversely 
as v when the temperature remains constant. 

This law was discovered in the seventeenth century by Robert Boyle. 
E.g., if the volume of a gas is 10 dm 3 under the ordinary pressure 
of the atmosphere (" under a pressure of one atmosphere "), it is 

as much when the pressure is n times as great, 

n times " " " " " " - " 

n 

the temperature always being considered constant. 

Example. A toy balloon contains 3 1 of gas when exposed to a 
pressure of 1 atmosphere ; what is its volume when the pressure is 
increased to 4 atmospheres ? decreased to of an atmosphere ? 

1. v the volume varies inversely as the pressure, it is i as much 
when the pressure is 4 times as great. 

2. Similarly, it is 8 times as much when the pressure is i as great. 

3. /. the volumes are 0.75 1 and 24 1. 

Exercises. 1 . Equal quantities of air are on opposite sides of a 
piston in a cylinder that is 12 in. long ; if the piston moves 3 in. from 
the center, find the ratios of the pressures. Draw the figure. 

2. If a cylinder of gas under a certain pressure has its volume 
increased from 20 1 to 25 1, what is the ratio of the pressures ? 

3. A cubic foot of air weighs 570 gr. at a pressure of 15 Ibs. to 
the square inch ; what will a cubic foot weigh at a pressure of 10 Ibs. ? 

4. A mass of air occupies 18 cu. in. under a pressure of 1\ Ibs. 
to the square inch; what space will it occupy under a pressure of 
25 Ibs. ? 

5. If the volume of a gas varies inversely as the height of the mer- 
cury in a barometer, and if a certain mass occupies 23 cu. in. when 
the barometer indicates 29.3 in., what will it occupy when the 
barometer indicates 30.7 in. ? 

6. A liter of air under ordinary pressure weighs 1.293 g when the 
barometer stands at 76 cm ; find the weight when the barometer stands 
at 82 cm, the weight varying inversely as the height of the barometer. 

7. A certain gas has a volume of 1200 cm 3 under a pressure of 
1033 g to 1 cm 2 ; find the volume when the pressure is 1250 g. 



HIGHER ARITHMETIC. 



II. PROPORTION. 



A proportion is an expression of equality of ratios. 



Thus, | = |, H = ^= ^, $3.50 : $7 = 4 books : 8 books, 3/4 = 

$6/$8, are examples of proportion. There is another symbol formerly 
much used to express the equality of ratios, the double colon (: :). 

There may be an equality of several ratios, as 1:2 = 4:8 
= 9 : 18, the term continued proportion being applied to such 
an expression. There may also be an equality between the 
products of ratios, as f f = \ - *-, such an expression 
being called a compound proportion. 

Arithmetic uses continued proportion but little, and problems in 
compound proportion are more easily solved by the unitary analysis. 

In the proportion a:b = c:d, a, b, c, d are called the 
terms, a and d the extremes, and b and c the means. 

If three of the four terms of a proportion are known, the other can 
be readily found by multiplying or dividing equals by equals. Thus, 

x 4 7 *4 

if - = - then x = -r- = 3.5. If this should seem to require multi- 
7 o o 

plying by a concrete number, the difficulty may be avoided by making 

$4 4 
each term of either ratio abstract; this is permissible because S = ' 

Exercises. 1. Given r = -> the terms being abstract numbers, to 
prove that the product of the means equals the product of the extremes. 

2. In the proportion - = - prove that x = bc/d ; that is, that one 

extreme equals the product of the means divided by the other extreme. 

3. In which of these proportions is the value of x the more easily 

. x 17 23 5 
found, and why? - = -,_ = -. 

x 7 

4. Find the value of x, correct to 0.1, in the proportions - = 

z:3 = 4:5, 0.3:7 =x : 1.52, ^ = y' 

5. Also in the proportions x : 1.273 = 0.4 : 2.3, 1.7 : 3 = x : 7. 



PROPORTION. 97 

If one quantity varies directly as another, the two are 
said to be directly proportional, or simply proportional. 

E.g., at retail the cost of a given quality of sugar varies directly 
as the weight ; the cost is then proportional to the weight. Thus, at 
4 cts. a pound 12 Ibs. cost 48 cts., and 4 cts. : 48 cts. = 1 Ib. : 12 Ibs. 

If one quantity varies inversely as another, the two are 
said to be inversely proportional. 

E.g., in general, the temperature being constant, the volume of a 
gas varies inversely as the pressure, and the volume is therefore said 
to be inversely proportional to the pressure. 

Exercises. 1. State which of the following, other things being 
equal, are directly proportional and which are inversely proportional : 
(a) Volume of gas, pressure. 
(6) Volume of gas, temperature. 

(c) Distance from fulcrum, weight. 

(d) Cost of carrying goods, distance. 

(e) Weight of goods carried for a given sum, distance. 
(/) Amount of work done, number of workers. 

(g) Price of bread, price of wheat. 

2. Given 1.43 : x = 4.01 : 2, find, correct to 0.01, the value of x. 

. x 63 

3. Also in - = 

7 x 

4. Also in 27 : x = x : 48. 

The applications of proportion are found chiefly in 
geometry and physics. While several types of business 
problems were formerly solved by this means, other 
methods are now generally employed. 

In the two illustrative examples on p. 98, the first three 
steps are explanatory of the statement of the proportion 
and may be omitted in practice. In the first problem 
the ratios are written in the fractional form in order that 
the reasons involved may appear more readily. The 
symbol for the unknown quantity may be placed in any 
term and the proportion arranged accordingly ; but if the 
solution is to be explained, it will be found more con- 
venient to place the x in the first term. (See p. 96, Ex. 3.) 



98 



HIGHER ARITHMETIC. 



Examples, (a) The time of oscillation of a pendulum is propor- 
tional to the square root of the number representing its length ; the 
length of a 1-sec. pendulum being 39.2 in., what is the length of a 
2-sec. pendulum ? 

1. Let x = the number of inches of length. 

2. Then ^r = the ratio of the lengths. 

o9.2 

3. And f = the ratio of the corresponding times of oscillations. . 

4. v the time is proportional to the square root of the number 
representing the length, 



. 

V39.2 

5. ... JL- = i, whence x = 39.2 4 =' 156.8. 

o9.2 

6. v x = the number of inches, .. the pendulum is 156.8 in. long. 

(b) A mass of air fills 10 dm 3 under a pressure of 3 kg to 1 cm 2 ; 
what is the space occupied under a pressure of 5 kg to 1 cm 2 , the 
temperature remaining constant ? (See Boyle's law, p. 95.) 

1. Let x the number of dm 3 under a pressure of 5 kg to 1 cm 2 . 

2. Then x : 10 = the ratio of the volumes. 

3. And 5:3 = the ratio of the corresponding pressures. 

4. v the volume is inversely proportional to the pressure, 

.-. x : 10 = 3 : 5. 

5. .-. = 10-3 : 5 = 6. 

6. v x = the number of dm 3 , .-. the space is 6 dm 3 . 



Exercises. 1. The distance through which a body falls from a 
state of rest is proportional to the square of the number representing 
the time of fall ; if a body falls 176.5 m in 6 sees., how far does it fall 
in 3.25 sees. ? 

2. Also in 1 sec. ? in 2 sees. ? in 4 sees. ? 

3. It is proved in mechanics that, neglecting friction, the power 

acting parallel to an inclined plane and neces- 
sary to support a weight is to that weight as 
the height to which the weight is raised is to 
the length of the incline. (In the figure, p : w 
= h : I = h': I'.) If the height is of the length, 
what power will support a 20-lb. weight (neg- 
lecting friction in this and similar problems) ? 




FIG. 18. 



PROPORTION. 99 

4. A train weighing 126 tons rests on an incline and is kept from 
moving down by a force of 1500 Ibs. ; the road rises 1 ft. in how many 
feet of its length ? 

5. On a plane rising 3 ft. in every 5 ft. of its length, how many 
pounds of force exerted parallel to the plane will keep a mass of 10 Ibs. 
from sliding ? 

6. What must be the length of an inclined plane in order that a 
man may roll a 500-lb. cask into a wagon 3.5 ft. high by the exertion 
of a force of 350 Ibs. ? 

7. How long is a pendulum which oscillates 56 times a minute ? 

8. If a pipe 1.5 cm in diameter fills a reservoir in 3.25 mins., how 
long will it take a pipe 3 cm in diameter to fill it ? 

9. If a projectile 8.1 in. in length weighs 108 Ibs., what is the 
weight of a similar projectile 9.37 in. long ? (Answer to 0.1 Ib.) 

10. The masses of two solids remaining unchanged, their attraction 
for each other is inversely proportional to the square of the number 
representing the distance between their centers of gravity ; if at a 
distance of 2 m they are attracted by a force of 1 mg, what will be 
their attraction at a distance of 1 km ? 

11. A body weighs 15 Ibs. 5000 mi. from the earth's center (i.e., 
about 1000 mi. above the surface); how much will it weigh 4000 mi. 
from the center ? 

12. If a body weighs 1 kg at the level of the sea, how much will it 
weigh at an elevation of 1 km ? (Take the radius as 6370 km ; give 
the result correct to 0.001 kg.) 

13. What is the height of a tower which casts a shadow 143 ft. 
long at the same time that a post 4.5 ft. high casts a shadow 6.1 ft. 
long? 

14. Of two bottles of similar shape one is twice as high as the 
other ; the smaller holds 0.5 pt., how much does the larger hold ? 

15. The amount of light received on a given surface being inversely 
proportional to the square of the number representing its distance 
from the source of light, if the amount per sq. in. at a distance of 1 ft. 
is represented by 1, what will represent the amount at a distance of 
0.1 in. ? (Answer to 0.01.) 

16. If the interest received on a certain sum for 1.5 yrs. is $27.50, 
how much is the interest on the same sum at the same rate for 
2 mo.? 

17. If the interest received on a certain sum for a certain time is 
$53 at 6 cts. for every dollar, how much is the interest on that sum 
for the same time at 4 cts. for every dollar ? 



100 HIGHER ARITHMETIC. 

18. When the barometer stands at 30 in., the pressure of the atmos- 
phere is 14.7 Ibs. per sq. in. ; what is the pressure when the barometer 
stands at 28 in. ? 

19. What is the width of a stream if a pole 64 ft. high 9 ft. from 
its bank casts a shadow which just reaches across the stream and the 
shadow of a nail in the pole 8 ft. from the ground just touches the 
bank? 

20. A water tower 160 ft. high and 35 ft. in diameter is to be 
represented in a drawing as 10 in. high ; how many inches in the 
representation of the diameter ? 

21. The Washington monument casts a shadow 223 ft. 6.5 in. when 
a post 3 ft. high casts a shadow 14.5 in. ; find the height of the monu- 
ment. 

22. If a triangle whose base is 2 in. long has an area of 3 sq. in., 
what is the area of a similar triangle whose corresponding base is 8 in. 
long? 

23. If a metal sphere 10 in. in diameter weighs 327.5 Ibs., what is 
the weight of a sphere of the same substance 14 in. in diameter ? 

24. A cube of water 1.8 dm on an edge weighs how many kg ? 

25. If a sphere whose surface is 16 it cm 2 weighs 5 kg, what 
is the weight of a sphere of the same substance whose surface is 
32 it cm 2 ? 

26. If the length of a 1-sec. pendulum be considered as 1 m, 
what is the time of oscillation of a pendulum 6.4 m long? 62.5 m 
long? 

27. On a map constructed on a scale of T ^oWo tne distance from 
Detroit to Chicago is 112.86 in. ; how many miles between these cities ? 

28. The ratio of immigrants from the United Kingdom to those 
from the rest of Europe during the decade from 1881 was 5 : 16 ; the 
total number from these two sources was 6,192,000 ; how many from 
each ? (Answer correct to 1000.) 

29. Kepler showed that the squares of the numbers representing 
the times of revolution of the planets about the sun are proportional 
to the cubes of the numbers representing their distances from the 
sun. Mars being 1.52369 as far as the earth from the sun, and the 
time of revolution of the earth being 365.256 da., find the time of 
revolution of Mars. 

30. Similarly, find the time of revolution of each of the following 
planets, the numbers representing relative distances from the sun, the 
distance to the earth being taken as the unit: Mercury 0.39, Venus 
0.72, Jupiter 5.20, Saturn 9.54, Uranus 19.18, Neptune 30.07. 



PROPORTION. 



101 



Problems in electricity. The great advance in elec- 
tricity in recent years renders necessary a knowledge of 
such technical terms as are in everyday use. Problems 
involving these terms belong to proportion, but may be 
omitted without interfering with the subsequent work. 

When water flows through a When electricity flows through 

pipe some resistance is offered a wire some resistance is offered, 
due to friction or other impedi- This resistance is measured in 
ment to the flow of the water. ohms. An ohm is the resistance 

offered by a column of mercury 
1 mm 2 in cross section, 106 cm 
long, at C. 



A certain quantity of water 
flows through the pipe in a second, 
and this may be stated in gallons 
or cubic inches, etc. 



A certain pressure is necessary 
to force the water through the 
pipe. This pressure may be meas- 
ured in pounds per sq. in., kilo- 
grams per cm 2 , etc. 



Hence, in considering the water 
necessary to do a certain amount 
of work (as to turn a water-wheel) 
it is necessary to consider not 
merely the pressure, for a little 
water may come from a great 
height, nor merely the volume, 
nor merely the resistance of the 
pipe; all three must be consid- 
ered. 



A certain quantity of electric- 
ity flows through the wire. This 
quantity is measured in amperes. 
An ampere is the current neces- 
sary to deposit 0.001118 g of silver 
a second in passing through a cer- 
tain solution of nitrate of silver. 

A certain pressure is necessary 
to force the electricity through the 
wire. This pressure is measured 
in volts. A volt is the pressure 
necessary to force 1 ampere 
through 1 ohm of resistance. 

Hence, in considering the elec- 
tricity necessary to do certain 
work it is necessary to consider 
not merely the voltage, for a little 
electricity may come with a high 
pressure, nor merely the amper- 
age, nor merely the number of 
ohms of resistance ; all three must 
be considered. 



The names of the electrical units mentioned come from the names 
of three eminent electricians, Ohm, Ampere, and Volta. 



If) 2 JSGHEK ARITHMETIC. 

It is proved in physics that the resistance of a wire 
varies directly as its length and inversely as the area of a 
cross section. 

That is, if a mile of a certain wire has a resistance of 3.58 ohms, 
2 mi. of that wire will have a resistance of 2-3.58 ohms, or 7.16 ohms. 
Also, 1 mi. of wire of the same material but of twice the sectional 
area will have a resistance of $ of 3.58 ohms, or 1.79 ohms. 

From these laws and definitions, the most common prob- 
lems and statements concerning electrical measurements 
will be understood. The student should not feel obliged, 
however, to use the proportion form in the solutions. 
Ordinary analysis, the unitary analysis, or the equation 
may be employed. 

Exercises. 1. If the resistance of 1 mi. of a certain electric-light 
wire is 3.58 ohms, what is the resistance of 5 mi. of wire of the same 
material but of twice the sectional area ? Also of 1 mi. of wire of the 
same material but of twice the diameter ? 

2. If the resistance of 700 yds. of a certain- cable is 0.91 ohm, what 
is the resistance of 1 mi. of that cable ? 

3. If the resistance of 100 yds. of a certain wire is 5 ohms, what 
length of the same wire would have a resistance of 13.2 ohms ? 

4. The resistance of a certain wire is 9.1 ohms, and the resistance 
of 1 mi. of this wire is known to be 1.3 ohms ; required the length of 
the wire. 

5. If the resistance of 130 yds. of copper wire y 1 ^ in. in diameter is 
1 ohm, what is the resistance of 100 yds. of copper wire -fa in. in 
diameter ? 

6. What is the resistance of 1 mi. of copper wire 1.14 mm in diam- 
eter, if the resistance of 1 mi. of copper wire 1.4 mm in diameter is 
8.29 ohms ? 

7. What is the length of copper wire 1 mm in diameter which has 
the same resistance as 6 m of copper wire 0.74 mm in diameter ? 

8. What must be the length of an iron wire of sectional area 4 mm 2 
to have the same resistance as a wire of pure copper 1000 m long 
whose sectional area is 1 mm 2 , taking the conductivity of iron to be \ 
of that of copper ? 



PROPORTION. 103 

9. How thick must an iron wire be in order that for the same length 
it shall offer the same resistance as a copper wire 2.5 mm in diameter ? 
(Answer to 0.01 mm.) 

10. If the resistance per cm 8 of a certain metal is 1.356 10~ 5 ohm, 
what is the resistance of a wire of this metal 1 m long and 2 mm thick ? 
(Answer to 10~ 5 ohm.) 

11. What is the ratio of the resistances of two wires of the same 
metal, one of which is 30.48 cm long and weighs 35 g, while the other 
is 18.29 cm long and weighs 10.5 g ? 

12. The resistance of 1 m of pure copper wire 1 mm in diameter 
is 0.02 ohm, and the resistance of a certain specimen of copper wire 
3 mm in diameter and 10 m long is 0.025 ; what is the ratio of their 
resistances ? 

13. The resistance of 1 mi. of a certain grade of copper wire whose 
diameter is 0.065 in. is 15.73 ohms, and the resistance of a wire of 
pure copper 1 ft. long and 0.001 in. in diameter is 9.94 ohms ; what is 
the ratio of their conductivities ? 

14. The resistance of a certain dynamo is 10.9 ohms and the resist- 
ance of the rest of the circuit is 73 ohms ; the electro-motive force of 
the machine being 839 volts, find how many amperes flow through the 
circuit. 

v 1 volt forces 1 ampere through 1 ohm of resistance, 839 volts will 

839 amperes 
force 839 amperes through 1 ohm of resistance, or 

through (73 + 10.9) ohms of resistance. 

15. The resistance of a dynamo being 1.6 ohms and the resistance 
of the rest of the circuit being 25.4 ohms, and the electro-motive force 
being 206 volts ; find how many amperes flow through the circuit. 

16. Three arc lamps on a circuit have a resistance of 3.12 ohms 
each; the resistance of the wires is 1.1 ohms, and that of the dynamo 
is 2.8 ohms ; find the voltage necessary to produce a current of 14.8 
amperes. 

17. Three arc lamps on a circuit have a resistance of 2.5 ohms each ; 
the resistance of the wires is 0.5 ohm, and that of the dynamo is 0.5 
ohm ; find the voltage necessary to produce a current of 25 amperes 
through the circuit. 

18. The resistance of a certain electric lamp is 3.8 ohms when a 
current of 10 amperes is flowing through it ; what is the voltage ? 

19. If 31.1 volts force 35.8 amperes through a lamp, what is the 
resistance ? 



CHAPTER X. 
Series. 



A series is a succession of terms formed according to 
some common law. 

E.g., in the following, each term is formed from the preceding as 
indicated : 

1,3,5,7, , by adding 2; 

7, 3, 1, 5, , by subtracting 4, or by adding 4; 

3, 9, 27, 81, , by multiplying by 3, or by dividing by ; 

64, 16, 4, 1, , , by dividing by 4, or by multiplying by i; 

2, 2, 2, 2, , by adding 0, or by multiplying by 1. 

In the series 0, 1, 1, 2, 3, 5, 8, 13, , each term after the first 

two is found by adding the two preceding terms. 

It is evident that the number of kinds of series is unlimited. 

An arithmetic series (also called an arithmetic progres- 
sion) is a series in which each term after the first is found 
by adding a constant to the preceding term. 

E.g., 7, 1, 5, 11, , the constant being 6, 

2, 2, 2, 2, , " " " 0, 

98, 66, 34, 2, , " " " -32. 

A geometric series (also called a geometric progression) 
is a series in which each term after the first is formed by 
multiplying the preceding term by a constant. 

E.g., 2, 5, 12i, 31 J, , the constant being 2-, 

3, -6, +12, -24, , " -2, 

10, 5, 2i, 1, , " " " i, 

2, 2, 2, 2, , " " " 1. 



SERIES. 105 

By custom, and because of their simplicity, arithmetic considers 
only arithmetic and geometric series. It should be stated, however, 
that the presence of this subject in applied arithmetic is merely a 
matter of tradition. It properly belongs to algebra, and hence may 
be omitted from the present course if Chap. XI is not taken. Since 
its application to business and to elementary science is slight, only a 
few of the more simple cases are considered. 

I. ARITHMETIC SERIES. 

Symbols. The following are in common use : 

n, the number of terms of the series. 
s, " sum " " " " 
ti, 2 , 3 , ..... t n , the terms of the series. 

In particular, a, or t\, the 1st term, and Z, or n , the nth or last term. 
d, the constant which added to any term gives the next ; d is usually 
called the difference. 

Formulae. There are two formulae in arithmetic series 
of such importance as to be designated as fundamental. 

1. t n , or I = a + (n 1) d. 

Proof. 1. t 2 = a + d by definition. 

t s = t 2 + d = a + 2 d. 

ti t s + d = a + 3 d. 

2. .-. 4 = t n -i + d = a + (n - l)d. 

3. Or I = a + (n - 1) d. 






Proof. 1. s = a + (a + d) + (a + 2 d) + ..... (I d) + 1. 

2. Hence, s = I + (I d) + (I 2 d)+ ..... (a + d) + a, by revers- 
ing the order. 

3. .-. 2 s (a + l) + (a + I) + (a + J), by adding equations 
1 and 2. 

4. .'.2s = n(a + l), v there is an (a+ 1) in step 3 for each 
of the n terms in step 1. 



5. ... ias - 



106 HIGHER ARITHMETIC. 

It is evident that from formulae 1 and 2, various others 
can easily be obtained. 

E.g., from I = a + (n 1) d it follows that 

I - (n - 1) d = a, ^y = d, etc. 

From s = ^-r - it follows that 

2s 

- - = n, etc. 

From I = a + (n 1) d and s *-r - it follows that 

_ n [a + a + (n - 1) d] 
S ~~ ~~2~ 



Exercises. 1. From s = *-r - i find a in terms of s, n, I. 

2. From I = a + (n 1) d, find n in terms of I, a, d. 

3. Find t w in the series 540, 480, 420, 

4. Find 100 in the series 1, 3, 5, 

5. Find s, given a = 1, I = 100, n = 100. 

6. Find s, given a = 10, n = 6, d = 4. Write out the series. 

7. Find s, given a = 40, n = 113, d = 5. 

8. Find n, given s = 36,160, a = 40, J = 600. 

9. What is the sum of the first 50 odd numbers ? the first 100 ? 
the first n ? 

10. What is the sum of the first 50 even numbers ? the first 100 ? 
the first n ? 

11. What is the sum of the first 100 numbers divisible by 5 ? by 7 ? 

12. $100 is placed at interest annually on the first of each January 
for 10 yrs. , at 6% ; find the total amount of principals and interest 
at the end of 10 yrs. 

13. A body falling in vacuum would fall 4.9 m in the first second, 
and in each succeeding second it would fall 9.8 m farther than in the 
preceding second ; how far would it fall in the llth second ? the 17th ? 

14. From the data of Ex. 13, how far would it fall in 11 sees. ? 
17 sees. ? 57 sees. ? 

15. How long has a body been falling when it passes through 
53.9 m during the last second ? 



SERIES. 107 



II. GEOMETRIC SERIES. 

Symbols. The following are in common use : 

?i, s, a, Z, and t\, t 2j ..... t n , as in arithmetic series ; 
r, the constant by which any term may be multiplied to produce 
the next ; r is usually called the rate or ratio. 

Formulae. There are two formulae in geometric series 
of such importance as to be designated as fundamental. 
or l = ar n ~\ 

Proof. 1. t 2 = ar by definition. 

ts = t^r = ar 2 . 
4 = t s r = ar 3 . 

2. .-. 4 = 4-ir = ar"- 1 . 

3. Or I = ar"- 1 . 



_ 
s 



n a Ir a 



Proof. 1. s = a + ar + ar 2 + ..... + ar"~ 2 + ar"- 1 . 

2. .-. rs = ar + ar 2 + ..... + ar"- 2 + ar"- 1 + ar n , by 
multiplying by r. 

3. .-. rs s = ar" a, by subtracting, (2) (1). 

4. .-. (r l)s = ar" a, and s = :- , by dividing by (r 1). 

5. And v ar n = ar"~ J r = Zr, .-. s = --- 

Exercises. 1. From I = ar"- 1 , find a in terms of J, n, r. 

2. Also r in terms of w, Z, a. 

3. From s = r- find a in terms of r, n, s. 

4. Find n in the series 1, 2, 4, 8, ..... ; also in the series 1, , 
i, ..... 

5. Find s, given a = 2, r = 4, w = 3; also, given t s = 50, r = 5, 
n =6. 

6. To what sum will $1 amount, at 4% compound interest, in 
5 yrs. ? (Here o = $l, r = 1.04, n = 6.) 

7. To what sum will $1 amount in 5 yrs. , at 4% a year, compounded 
semi-annually ? 



108 HIGHER ARITHMETIC. 

If the number of terms is infinite and r < 1, then s 
approaches as its limit 

This is indicated by the symbols s = _ n being infinite. 

The symbol == is read "approaches as its limit." 
Proof. 1. v r<l, the terms are becoming smaller, each being 
divided to obtain the next. 

2. .-. I = 0, and .-. Ir = 0, although they never reach that limit. 

3. .-. s = _ i by formula 2. 

4. .-. s = _ , by multiplying each term of the fraction by 1. 
E.g., consider the series 1, |, J, , where n is infinite. Here, 

s = , or -, or 2. That is, the greater the number of terms, 

i r l i 

the nearer the sum approaches 2, although it never reaches it for 
finite values of n. 

Exercises. 1. Find the limits of the following sums, n being 
infinite : 

(a) 10 + 5 + 2|+ (d) 90 + 9 + 0.9 + 0.09+ 

(6) 10+(-5)+2i+(-l) + (e) i + i + sV+ 

(c) 1 + 0.1 + 0.01 + - (/) i + + T fr + 

2. Given s == 2, r = J, find a. 

3. Given s = 4f , a = 4, find r. 

4. Given s = 1, r = fff , find a. 

5. Find the sum of the first 20 terms of the following series : 32, 
16, 8, 4, 

6. Given s = 155, r = 2, n = 5, find a. 

7. Given s = 124.4, r = 3, n = 4, find a. 

8. Suppose an elastic ball falls 10 ft. and rebounds half that 
distance, then falls 5 ft. and rebounds half that distance, and so on, 
rebounding half the distance fallen each time ; required the limit of 
the sum of the distances fallen ; of the distances rebounded ; of the 
total distances through which the ball passes. 

9. If a man invests $1 at the beginning of each year at 4% com- 
pound interest, what will be the sum of the principals and interest at 
the end of 5 yrs. ? 



SERIES. 109 

Circulating decimals. If the fraction f> T is reduced to 

the decimal form, the result is 0.272727 , and similarly 

the fraction 1^ = 0.152777 The former constantly 

repeats 27, and the latter constantly repeats 7 after 0.152. 

When, beginning with a certain order of a decimal frac- 
tion, the figures constantly repeat in the same order, the 
number is called a circulating, repeating, or recurring deci- 
mal, and the part which repeats is called a circulate, 
repetend, or recurring period. 

These various names are used, the subject being of too little prac- 
tical importance to establish a uniform custom. 

A circulate is usually represented by placing a dot over its first and 
last figures, thus : 

0.272727 is represented by 0.27 ; 

0.152777 " " " 0.1527. 

A circulating decimal may be reduced to a common 
fraction by means of the formula s = ^ , as in the 
following examples : 

1. To what common fraction is 0.27 equal ? 

1. 0.27 = 0.27 + 0.0027 + 0.000027 + 

2. This is a geometric series with a = 0.27, r = 0.01, n infinite. 

0.27 27 3 



1 - 0.01 99 11 
2. To what common fraction is 0.1527 equal ? 

1. 0.1527 = 0.152 + 0.0007 + 0.00007 + 

2. Of this, 0.0007 + 0.00007 + is a geometric series with 

a = 0.0007, r = 0.1, n infinite. 

. 0.0007 _ 7 
"1-0.1 ~9000' 
4. To this must be added 0.152, giving 0.152, or $j$, or \\. 

Exercise. Express as common fractions : 

(a) 0.9. (d) 2.476. 

(6) O.Oi. (e) 0.003714. 

(c) 0.247. (/) 0.123450. 



CHAPTER XL 
Logarithms. 



ABOUT the year 1614 a Scotchman, John Napier, 
invented a system by which multiplication can be per- 
formed by addition, division by subtraction, involution by 
a single multiplication, and evolution by a single division. 

This chapter might properly have followed the work on the four 
fundamental operations. By reserving it until this time, however, 
the practical application to scientific problems and the relation to 
series are more evident. It is not necessary for the understanding of 
the subsequent chapters and may, therefore, be omitted if desired. 
For the student who proposes to take even an elementary course in 
physics, however, the subject will be found of much value. 

In considering the annexed series 90 1 

of numbers it is apparent that, 2 1 2 2 7 = 128 

1. v 2 3 2 6 = 2 8 , 2 2 = 4 2 8 = 256 
... 8 32 = 2 8 = 256 ; 2 3 = 8 2 9 = 512 

.-. the product can be found by adding the 2* = 16 2 10 = 1024 
exponents (3 + 6 = 8) and then finding what 2 5 = 32 2 11 = 2048 
2 8 equals. 

2. v 2 9 : 2 3 = 2 6 , 

.-. 512 : 8 = 64 ; .-. this quotient can be found from the table by 
a single subtraction of exponents. 

3. v (25)2 = 2 5 2 5 = 2 10 , 
.-. 32 2 - 1024. 

4. v V2 = \/2 5 2 5 = 2 s , 
.-. V1024 = 32. 

5. The exponents of 2 form an arithmetic series, while the powers 
form a geometric series. 



LOGARITHMS. Ill 

In like manner a table of the powers of any number may 
be made and the four operations, multiplication, division, 
involution, evolution, reduced to the operations of addition, 
subtraction, multiplication, and division of exponents. 

For practical purposes, the exponents of the powers to 
which 10, the base of our system of counting, must be raised 
to produce various numbers are put in a table, and these 
exponents are called the logarithms of those numbers. 

In this connection the word "power" is used in its broadest sense, 
10 W being considered as a power whether n is positive, negative, inte- 
gral, or fractional. The logarithm of 100 is written "log 100." 
E.g., 10 3 = 1000, .-. log 1000 = 3. 10 = 1, .-. log 1 =0. 

102 = 10 Q, ... log 100 = 2. 10-i = _!_, ... i g o.l = - 1. 

101 = 10, .-. log 10 = 1. 10-2 = , .-. log 0.01 = - 2. 



, that is, the thousandth root of 10 801 , is nearly 2, 

.-. log 2 = 0.301 nearly. 

Although log 2 cannot be expressed exactly as a decimal fraction, 
it can be found to any required degree of accuracy. In the present 
work logarithms are given to 4 decimal places ; logarithms to 5 or 6 
decimal places are sufficient for ordinary computations of considerable 
length. 

Exercises. 1. What are the logarithms of these numbers ? 
(a) 100,000; (6) T oV<j, or 0.001; (c) 10 7 ; (d) 0.00001; (e) 10~ 6 ; 
(/) VlO, or 10*; (g) -fao. 

2. What is meant by saying that the logarithm of 50 is 1.699 ? by 
saying that the logarithm of "300 is 2.4771 ? 

3. Between what two consecutive integers does log 500 lie, and 
why ? also log 2578 ? log 17 ? log 923,467 ? 

4. Between what two consecutive negative integers does log 0.02 
lie, and why ? also log 0.007 ? log 0.0009 ? log 0.025 ? 

5. What is the logarithm of 10* 10 6 ? of 10 9 : 10 3 ? of VlO~ 8 ? of 



6. What is the logarithm of 10 2 10 3 10 4 ? of 10 4 10 5 10 7 ? of 
0.001 of 10 2 10 3 ? 



7. What is the logarithm of VlO 4 10 6 10 8 ? of VlOi 5 10 20 ? of 

vTo? 



112 HIGHER ARITHMETIC. 

Since 2473 lies between 1000 and 10,000, its logarithm 
lies between 3 and 4. It has been computed to be 3.3932. 
The integral part 3 is called the characteristic of the loga- 
rithm, and the fractional part 0.3932 the mantissa. 

That is, 108, O r 10 3 - 3932 = 2473, .-. log 2473 = 3.3932. 

... 1Q3.3932 :ioi=10 2 - 3932 , .-. 10 2 - 3932 = 247.3, .. log 247.3 = 2.3932. 

Similarly, iQi.3m =24.73, .-. log 24.73 = 1.3932. 

10- 3932 =2.473, .-.log 2.473 = 0.3932. 

" 100.8932-1= 0.2473, .-. logO.2473 = 0.3932-1. 

It is thus seen that 

1. The characteristic can always be found by inspection. 

Thus, because 438 lies between 100 and 1000, hence log 438 lies 
between 2 and 3, and log 438 = 2 + some mantissa. 

Similarly, 0.0073 lies between 0.001 and 0.01, hence log 0.0073 lies 
between 3 and 2, and log 0.0073 = 3 + some mantissa. 

2. The mantissa is the same for any given succession of 
digits, wherever the decimal point may be. 

Thus, log 2473 = 3.3932, and log 0.2473 = 0.3932 - 1. 

3. Therefore, only the mantissas need be put in a table. 

Instead of writing the negative characteristic after the mantissa, 
it is often written before it, but with a minus sign above ; thus, log 
0.2473 = 0.3932 1 = T.3932, this meaning that only the character- 
istic is negative, the mantissa remaining positive. 

Negative numbers are not considered as having logarithms, but 
operations involving negative numbers are easily performed. E.g., 
the multiplication expressed by 1.478 ( 0.007283) is performed as 
if the numbers were positive, and the proper sign is prefixed. 

Exercises. 1. What is the characteristic of the logarithm of a 
number of 3 integral places ? of 5 ? of 10 ? ofn? 

2. What is the characteristic of the logarithm of 0.2? of any 
decimal fraction whose first significant figure is in the first decimal 
place ? the second decimal place ? the 10th ? the nth ? 

3. From Exs. 1, 2 formulate a rule for determining the character- 
istic of the logarithm of any positive number. 

4. If log 39,703 = 4.5988, what are the logarithms of (a) 39,703,000, 
(6) 397.03, (c) 3.9703, (d) 0.00039703 ? 



LOGARITHMS. 113 

The fundamental theorems of logarithms. 
I. The logarithm of the product of two numbers equals 
the sum of their logarithms. 

1. Let a = 10, then log a = m. 

2. Let b = 10 n , " log b=n. 

3. .-. ab = 10 m+n , and log ab = m-\- n = log a + log b. 

II. The logarithm of the quotient of two numbers equals 
the logarithm of the dividend minus the logarithm of the 
divisor. 

1. Let a = 10, then log a = m. 

2. Let b = 10", log b = n. 

3..'. = i = 10 , and l og = m-rc. 



III. TAe logarithm of the nth power of a number equals 
n times the logarithm of the number. 

1. Let a = 10 m , then log a = w. 

2. /. a B = 10 BW , and log a n = ram = n log a. 

IV. The logarithm of the nth root of a number equals 

- th of the logarithm of the number. 



1. Leta = 10 w , then log a = m. 



m 



- 

2. /. a" = 10", and log a n = = - -log a. 

n n 

Th. Ill might have been stated more generally, so as to include 

- x 
Th. IV, thus : Log a y = - log a. The proof would be substantially 

the same as in Ths. Ill and IV. 

Exercises. Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, 
log 7 = 0.8451, and log 514 = 2.7110, find the following : 

1. log 6. 2. log 14. 3. Iog7 10 . 4. logV2. 

5. log 42. 6. log 5*. 7. log 105. 8. log 1.05. 

9. logV514. 10. Iog514 2 . 11. log 1542. 12. log 257. 

13. log 1799 [= log (| -514 -7)]. 14. log A/3*. 15. log V21. 

16. Show how to find log 5, given log 2. 



114 



HIGHER AKITHMETIC. 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 





0000 


0000 


3010 


4771 


6021 


6990 


7782 


8451 


9031 


9542 


i 


0000 


0414 


0792 


1139 


1461 


1761 


2041 


2304 


2553 


27S8 


2 


3010 


3222 


3424 


3617 


3802 


3979 


4150 


4314 


4472 


4624 


3 


4771 


4914 


5051 


5185 


5315 


5441 


5563 


5682 


5798 


5911 


4 


6021 


6128 


6232 


6335 


6435 


6532 


6628 


6721 


6812 


6902 


5 


6990 


7076 


7160 


7243 


7324 


7404 


7482 


7559 


7634 


7709 


6 


7782 


7853 


7924 


7993 


8062 


8129 


8195 


8261 


8325 


8388 


7 


8451 


8513 


8573 


8633 


8692 


8751 


8808 


8865 


8921 


8976 


8 


9031 


9085 


9138 


9191 


9243 


9294 


9345 


9395 


9445 


9494 


9 


9542 


9590 


9638 


9685 


9731 


9777 


9823 


9868 


9912 


9956 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


11 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 


0719 


0755 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


13 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


14 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 


15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


17 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 


2504 


2529 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


19 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


21 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 


3385 


3404 


22 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 


3579 


3598 


23 


3617 


3636 


3655 


3674 


3692 


3711 


3729 


3747 


3766 


3784 


24 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3945 


3962 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


27 


4314 


4330 


4346 


4362 


4378 


4393 


4409 


4425 


4440 


4456 


28 


4472 


4487 


4502 


4518 


4533 


4548 


4564 


4579 


4594 


4609 


29 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


5011 


5024 


5038 


32 


5051 


5065 


5079 


5092 


5105 


5119 


5132 


5145 


5159 


5172 


33 


5185 


5198 


5211 


5224 


5237 


5250 


5263 


5276 


5289 


5302 


34 


5315 


5328 


5340 


5353 


5366 


5378 


5391 


5403 


5416 


5428 


35 


5441 


5453 


5465 


5478 


5490 


5502 


5514 


5527 


5539 


5551 


36 


5563 


5575 


5587 


5599 


5611 


5623 


5635 


5647 


5658 


5670 


37 


5682 


5694 


5705 


5717 


5729 


5740 


5752 


5763 


5775 


5786 


38 


5798 


5809 


5821 


5832 


5843 


5855 


5866 


5877 


5888 


5899 


39 


5911 


5922 


5933 


5944 


5955 


5966 


5977 


5988 


5999 


6010 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


41 


6128 


6138 


6149 


6160 


6170 


6180 


6191 


6201 


6212 


6222 


42 


6232 


6243 


6253 


6263 


6274 


6284 


6294 


6304 


6314 


6325 


43 


6335 


6345 


6355 


6365 


6375 


6385 


6395 


6405 


6415 


6425 


44 


6435 


6444 


6454 


6464 


6474 


6484 


6493 


6503 


6513 


6522 


45 


6532 


6542 


6551 


6561 


6571 


6580 


6590 


6599 


6609 


6618 


46 


6628 


6637 


6646 


6656 


6665 


6675 


6684 


6693 


6702 


6712 


47 


6721 


6730 


6739 


6749 


6758 


6767 


6776 


6785 


6794 


6803 


48 


6812 


6821 


6830 


6839 


6848 


6857 


6866 


6875 


6884 


6893 


49 


6902 


6911 


6920 


6928 


6937 


6946 


6955 


6964 


6972 


6981 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



LOGARITHMS. 



115 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 


51 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7143 


7152 


52 


7160 


7168 


7177 


7185 


7193 


7202 


7210 


7218 


7226 


7235 


53 


7243 


7251 


7259 


7267 


7275 


7284 


7292 


7300 


7308 


7316 


54 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7396 


55 


7404 


7412 


7419 


7427 


7435 


7443 


7451 


7459 


7466 


7474 


56 


7482 


7490 


7497 


7505 


7513 


7520 


7528 


7536 


7543 


7551 


57 


7559 


7566 


7574 


7582 


7589 


7597 


7604 


7612 


7619 


7627 


58 


7634 


7642 


7649 


7657 


7664 


7672 


7679 


7686 


7694 


7701 


59 


7709 


7716 


7723 


7731 


7738 


7745 


7752 


7760 


7767 


7774 


60 


7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7846 


61 


7853 


7860 


7868 


7875 


7882 


7889 


7896 


7903 


7910 


7917 


62 


7924 


7931 


7938 


7945 


7952 


7959 


7966 


7973 


7980 


7987 


63 


7993 


8000 


8007 


8014 


8021 


8028 


8035 


8041 


8048 


8055 


61 


8062 


8069 


8075 


8082 


8089 


8096 


8102 


8109 


8116 


8122 


65 


8129 


8136 


8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 


66 


8195 


8202 


8209 


8215 


8222 


8228 


8235 


8241 


8248 


8254 


67 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8306 


8312 


8319 


68 


8325 


8331 


8338 


8344 


8351 


8357 


8363 


8370 


8376 


8382 


69 


8388 


8395 


8401 


8407 


8414 


8420 


8426 


8432 


8439 


8445 


70 


8451 


8457 


8463 


8470 


8476 


8482 


8488 


8494 


8500 


8506 


71 


8513 


8519 


8525 


8531 


8537 


8543 


8549 


8555 


8561 


8567 


72 


8573 


8579 


8585 


8591 


8597 


8603 


8609 


8615 


8621 


8627 


73 


8633 


8639 


8645 


8651 


8657 


8663 


8669 


8675 


8681 


8686 


74 


8692 


8698 


8704 


8710 


8716 


8722 


8727 


8733 


8739 


8745 


75 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 


76 


8808 


8814 


8820 


8825 


8831 


8837 


8842 


8848 


8854 


8859 


77 


8865 


8871 


8876 


8882 


8887 


8893 


8899 


8904 


8910 


8915 


78 


8921 


8927 


8932 


8938 


8943 


8949 


8954 


8960 


8965 


8971 


79 


8976 


8982 


8987 


8993 


8998 


9004 


9009 


9015 


9020 


9025 


80 


9031 


9036 


9042 


9047 


9053 


9058 


9063 


9069 


9074 


9079 


81 


9085 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9133 


82 


9138 


9143 


9149 


9154 


9159 


9165 


9170 


9175 


9180 


9186 


83 


9191 


9196 


9201 


9206 


9212 


9217 


9222 


9227 


9232 


9238 


84 


9243 


9248 


9253 


9258 


9263 


9269 


9274 


9279 


9284 


9289 


85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 


86 


9345 


9350 


9355 


9360 


9365 


9370 


9375 


9380 


9385 


9390 


87 


9395 


9400 


9405 


9410 


9415 


9420 


9425 


9430 


9435 


9440 


88 


9445 


9450 


9455 


9460 


9465 


9469 


9474 


9479 


9484 


9489 


89 


9494 


9499 


9504 


9509 


9513 


9518 


9523 


9528 


9533 


9538 


90 


9542 


9547 


9552 


9557 


9562 


9566 


9571 


9576 


9581 


9586 


91 


9590 


9595 


9600 


9605 


9609 


9614 


9619 


9624 


9628 


9633 


92 


9638 


9643 


9647 


9652 


9657 


9661 


9666 


9671 


9675 


9680 


93 


9685 


9689 


9694 


9699 


9703 


9708 


9713 


9717 


9722 


9727 


94 


9731 


9736 


9741 


9745 


9750 


9754 


9759 


9763 


9768 


9773 


95 


9777 


9782 


9786 


9791 


9795 


9800 


9805 


9809 


9814 


9818 


96 


9823 


9827 


9832 


9836 


9841 


9845 


9850 


9854 


9859 


9863 


97 


9868 


9872 


9877 


9881 


9886 


9890 


9894 


9899 


9903 


9908 


98 


9912 


9917 


9921 


9926 


9930 


9934 


9939 


9943 


9948 


9952 


99 


9956 


9961 


9965 


9969 


9974 


9978 


9983 


9987 


9991 


9996 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 



116 



HIGHER ARITHMETIC. 



Explanation of table. Given a number to find its loga- 
rithm. In the table on pp. 114 and 115 only the mantissas 
are given. For example, in the row opposite 71, and 
under 0, 1, 2, will be found : 



N 
71 





1 


2 


3 


4 


5 


6 


7 


8 


9 


8513 


8519 


8525 


8531 


8537 


8543 


8549 


8555 


8561 


8567 



This means that the mantissa of log 710 is 0.8513, of 
log 711 it is 0.8519, and so on to log 719. Hence, 
log 716 = 2.8643, log 7.18 = 0.8661, 

log 71,600 = 4.8549, log 0.0719 = 2.8567. 
And v 7154 is T \ of the way from 7150 to 7160, .'. log 
7154 is about T ^ of the way from log 7150 to log 7160. 
.'. log 7154 log 7150 -\-j\ of the difference between 

log 7150 and log 7160 
= 3.8543 + T % of 0.0006 
= 3.8543 + 0.0002 = 3.8545. 

The above process of finding the logarithm of a number of four 
significant figures is called interpolation. It is merely an approxima- 
tion available within small limits, since numbers do not vary as their 
logarithms, the numbers forming a geometric series while the loga- 
rithms form an arithmetic series. It should be mentioned again that 
the mantissas given in the table are only approximate, being correct 
to 0.0001. This is far enough to give a result which is correct to 
three figures in general, and usually to four, an approximation suf- 
ficiently exact for many practical computations. 

In all work with logarithms, the characteristic should be written 
before the table is consulted, even if it is 0. Otherwise it is liable to 
be forgotten, in which case the computation will be valueless. 



Exercises. From the table find the following : 
1. log 38. 2. log 743. 3. log 14,000. 

5. log 3.81. 6. log 0.00123. 7. log 1855. 

9. log 1.823. 10. log 0.2769. 11. log 0.00001727. 



4. log 3940. 
8. log 23.41. 
logV4T28. 



12. 



13. Iog9.821 3 . 14. log 75.55*. 15. log 0.0129 5 . 16. logV125. 



LOGARITHMS. 



117 



Given a logarithm to find the corresponding number. The 
number to which a logarithm corresponds is called its anti- 
logarithm. 

E.g., v log 2 = 0.3010, .-. antilog 0.3010 = 2. 

The method of finding antilogarithms will be seen from 
a few illustrations. Referring again to the row after 71 
on p. 115, we have : 



N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


71 


8513 


8519 


8525 


8531 


8537 


8543 


8549 


8555 


8561 


8567 



Hence, we see that 

antilog 0.8513 = 7.1, antilog 5.8531 = 713,000, 

antilog 2.8567 = 0.0719, antilog 1.8555 = 0.717. 

Furthermore, v 8540 is % way from 8537 to 8543, 
/. antilog 2.8540 is about % way from antilog 2.8537 to 
antilog 2.8543. 

.'. antilog 2.8540 is about % way from 714 to 715. 
.'. antilog 2.8540 = 714.5. 
Similarly, to find antilog 1.8563. 

antilog 1.8567 = 0.719 1.8563 

antilog 1.8561 = 0.718 1.8561 

6 2 

.-. antilog 1.8563 = 0.718$ = 0.7183. 

The interpolation here explained is, as stated on p. 116, merely a 
close approximation ; it cannot be depended upon to give a result 
beyond four significant figures except when larger tables are employed. 

Exercises. From the table find the following : 



1. antilog 0.1234. 
4. antilog 4.2183. 
7. antilog 0.9485 3. 
10. antilog 0.6585 5. 
13. antilog 0.6120-1. 
16. antilog 0.9290 - 2. 


2. antilog 3.4271. 
5. antilog 1.9286. 
8. antilog 5.7834. 
11. antilog 10.5441. 
14. antilog 0.7070. 
17. antilog 3.8320. 


3. antilog 2.8193. 
6. antilog 1.7829. 
9. antilog 0.9996. 
12. antilog 2.0000. 
15. antilog 1.7850. 
18. antilog 3.6387. 



118 HIGHER ARITHMETIC. 

Cologarithms. In cases of division by a number n it is 

often more convenient to add the logarithm of - than to 

n 

subtract the logarithm of n. The logarithm of - is called 

n 

the cologarithm of n. 

v log - = log 1 log n = log n, 

/. colog n log n. 

Also, colog n = 10 log n 10, often a more convenient form 
to use. 

The cologarithm can evidently be found by subtracting 
each digit from 9, excepting the right-hand significant one 
(which must be subtracted from 10) and the zeros follow- 
ing, and then subtracting 10. 

E.g., to find colog 6178. 

9. 9 9 9 10 

log 6178 = 3. 7 9 9 

colog 6178 = 6. 2 9 1-10. 

To find colog 41.5. 

9. 9 9 10 

log 41.5 = 1. 6 1 8 

colog 41.5 = 8. 3 8 2 0-10. 

To find colog 0.013. 

9. 9 9 9 10 

log 0.013 = 2. 1 1 3 9 

colog 0.013 = 11. 8 8 6 1 - 10 = 1.8861. 

In case the characteristic exceeds 10 but is less than 20, colog n 
may be written 20 log n 20, and so for other cases ; but these 
cases are so rare that they may be neglected at this time. 

The advantage of using cologarithms will be apparent 
from a single example : 

317 92 



To find the value of 



6178-0.13 



LOGARITHMS. 119 

USING COLOGABITHMS. NOT USING COLOGAKITHMS. 

log 317 = 2.5011 log 317 = 2.5011 

log 92 = 1.9638 log 92 = 1.9638 

colog 6178 = 6.2091 - 10 log (317 -92) = 4.4649 

colog 0.13 = 10.8861-10 log 6178 - 3.7909 

log 36.32 = 1.5601 log 0.13 = 1.1139 

log (6178 -0.13) = 2.9048 

log (31 7 -92) = 4. 4649 

log (6178 -0.13) = 2.9048 

log 36.32 = 1.5601 

Computations by logarithms. A few illustrative prob- 
lems will now be given covering the types which the 
student will most frequently meet. It is urged that all 
work be neatly arranged, since as many errors arise 
from failure in this respect as from any other single 
cause. 

Since TT enters so frequently into computations, the 
following logarithms will be found useful : 

log TT = 0.4971, log i = T.5029. 

1. Find the value of |^| 

log 0.007 = 0.8451 - 3 

3 -log 0.007= 2.5353 9 

colog 0.03625 = 11.4407 10 

13.9760 - 19 

= 0.9760- 6 = log 0.000009462. 



It will be noticed that the negative characteristic is less confusing 
if written by itself at the right. 

2. Find the value of 0.09515*. 

log 0.09515 = 0.9784 - 2. 
v the characteristic ( 2) is not divisible by 3, this may be written 

log 0.09515 = 1.9784 - 3. 

Then fr log 0.09515 = 0.6595 - 1 = log 0.4566. 

.-. 0.4566 = Ans. 



120 HIGHER ARITHMETIC. 

2.706-0.3-0.001279 



3. Find the value of 



86090 



2 706 -3-1 279 

This may at once be written ZTlTT^r '10~ 8 , thus simplify- 
ing the characteristics. Then 

log 2. 706 = 0.4324 
log 3 = 0.4771 
log 1.279 = 0.1069 
colog 8.609 = 9.0650 10 

log 1.206 = 0.0814 
.-. 1.206 -10- 8 =Ans. 

4. Given 2* = 7, find x, the result to be correct to 0.01. 

x log 2 = log 7. 

_ log 7 _ 0.8451 _ 
~~ log 2 ~~ 0.3010 ~~ 

This division may be performed directly or by finding the anti- 
logarithm of (log 0.8451 log 0.3010), the former being the more 
expeditious in this case. 

5. The weight of an iron sphere, specific gravity 7.8, is 14.3 kg; 
find the radius. 

v = |- 7tr s 1 cm 3 = volume in cm 3 . 
.-. weight = | Tzrr 3 7.8 1 g = 14,300 g. 

' r = ( ~4 ^~TT ) > the number of centimeters of radius. 

\4i7t ' 7.8 / 

log 3 = 0.4771 

log 14,300 = 4.1553 

colog 4 = 9.3979 10 

colog it 9.5029 10 

colog 7.8 = 9.1079-10 



3 2.6411 



log 7.593 = 0.8804 .-. radius = 7.593 cm. 

Exercises. 1. Find the value of (12.8 -r 0.07235)*. 

2. Find the value of (42 9.37)* -f 0.127*. 

3. Find the value of (4.376/Tzr)*. 

4. Given x : 4.127 = 0.125 : 2736 ; find x. 

5. Given x 3 = x 5 : 5 ; find x. 

6. Find the value of 27^. 

7. Given 117,600 = 7"- 1 ; find n, correct to 0.1. 

8. Find the value of Vjr-4.927. 



LOGARITHMS. 121 

9. Given 0.47 : x = x : 1.249 ; find z. 

10. Find the value of 0.00234 72.28 5.126 -10-7. 

11. What power will just raise a weight of 17.5 Ibs., the fulcrum 
of the lever being 1.73 ft. from the weight and 4.19 ft. from the power ? 

12. At what distance from the fulcrum must a power of 91 Ibs. be 
exerted to raise a weight of 7493 Ibs. 2 ft. 3 in. from the fulcrum ? 

13. It is shown in studying the strength of materials that a cylin- 
drical iron shaft 5f in. in diameter and 5 ft. 7.5 in. long between its 
supports will support a load at the center of 0. 726 -d- 8700 Ibs. /, 
where d = the number of inches of diameter and I = the number of 
feet of length. Perform the computation. 

14. 2240 Ibs. of chalk occupy 15.5 cu. ft., and a cubic foot of water 
weighs 62 Ibs. ; what is the specific gravity of chalk ? 

15. The surface of a sphere is 4 sq. in. ; what is its volume ? 

16. The volume of a sphere is 1 cu. ft. ; what is its radius ? 

17. What is the specific gravity of a substance of which a sphere 
of radius 9 cm weighs 15 kg ? 

18. What is the weight of a silver cone of radius 2 cm and height 
3.6 cm, the specific gravity of silver being 10.5 ? 

19. If the intensity of light varies inversely as the square of the 
distance and directly as the illuminating power of its source, what is 
the ratio of the intensities of a candle 3.75 ft. distant and a 41.5 
candle lamp 13.2 ft. distant ? (Answer to 0.01.) 

20. If the intensity of the light of the full moon is found to be 
equal to that of a candle at a distance of 4 ft. , what is the equivalent 
in candle power of the moon ? (See Ex. 19 ; take the distance of the 
moon as 2.4 10 5 mi.) 

[Omit the following if the problems in electricity were not taken.] 

21. The resistance of 1 cm 3 of copper is 1.6-10- 8 ohm at 0. 
and the resistance increases by 3.9- 10 ~ 3 of this for each degree rise 
in temperature ; find the resistance of a wire 10 m long and 1 mm in 
diameter at 25 C. 

22. An incandescent lamp takes a current of 0.71 ampere and the 
electro-motive force is 98.5 volts; what is the resistance of three 
such lamps ? 

23. If an incandescent lamp of 81 ohms resistance takes a current 
of 0.756 ampere, what is the voltage ? 

24. A mile of telegraph wire 2 mm in diameter offers a resistance 
of 12.85 ohms ; what is the resistance of 439 yds. of wire of the same 
material 0.8 mm in diameter ? 



CHAPTER XII. 
Graphic Arithmetic. 



WHEN the mind seeks to clearly appreciate the relations 
between several measurements, it is of great value to resort 
to a graphic representation, accurately drawn to a scale. 
Thus, to say that the distance in millions of miles from the 
Sun to Mercury is 36, to Earth 92, to Jupiter 481, and to 
Neptune 2778 is not nearly as expressive as when accom- 
panied by the following graphic representation of these 
measurements : 



111 



20 



15 



10 






NO. OF 
REPORT! 
JUT OF 100 



230 



T 



The annexed curve 
has been plotted to 
represent graphi- 
cally the statistics 
compiled by a state 
board of health with 
reference to typhoid 
fever and the condi- 
tion of wells for the 
various months in 
the year. 

The dotted line 
shows the average 
number of inches above the surface of water in the wells, 



II 



, \ 



220 



200 



INCHES 
ABOVE 
WATER 



21. 



GRAPHIC ARITHMETIC. 



123 



and hence is highest when the water is lowest ; the black 
line shows the number of reports, out of every 100, in which 
typhoid fever was mentioned as prevalent. The effect of 
low water upon typhoid fever is thus much more clearly 
seen than it would be from the mere numbers. 



-$150,000,000 



-$100,000,000 



$50,000,000 



1861-2-3-4-5 



1890-1-2-3-4-5-6 



FIG. 22. 



The above curve represents the sums paid by the United 
States government for pensions in various years as follows 
(by millions of dollars) : 

1862, 0.8 ; 1863, 1 ; 1864, 5 ; 1865, 8 ; 1870, 28 ; 1880, 57 ; 1890, 106 ; 
1891, 119; 1892, 141; 1893, 158; 1894, 141; 1895, 141; 1896, 139. 

Exercises. 1. In cases of contagious diseases the premises should 
be isolated and disinfected. In a certain part of the country, where 
this was neglected, the number of cases and deaths to each outbreak 
of diphtheria averaged respectively 13.78 and 3.81 ; in the same state, 
but where these precautions were taken, the corresponding results 
were 2.45 and 0.69. Represent graphically by four straight lines. 

2. Represent graphically the following per capita indebtedness as 
given in recent government reports : Austria-Hungary $71, France 
$116, Prussia $37, Great Britain and Ireland $88, Italy $76, Russia 
$31, Spain $74, United States $15. 



124 



HIGHER ARITHMETIC. 



3. A foot-ton is the measure of force required to raise 1 ton 1 ft. 
Plot the five curves corresponding to the following statistics and show 
where the growth has been the most marked. 





MILLIONS OF FOOT -TONS DAILY. 


FOOT-TONS 


YEAR. 


HAND. 


HORSE. 


STEAM. 


TOTAL. 


INHABITANT. 


1820 


753 


3,300 


240 


4,293 


446 


1840 


1,406 


12,900 


3,040 


17,346 


1,020 


1860 


2,805 


22,200 


14,000 


39,005 


1,240 


1880 


4,450 


36,600 


36,340 


77,390 


1,545 


1885 


6,406 


55,200 


67,700 


129,306 


1,940 



4. Represent the following statistics graphically : 





MILLIONS OF FOOT-TONS DAILY. 


FOOT-TONS 
PER. IN- 
HABIT ANT. 


HAND. 


HORSE. 


STEAM. 


TOTAL. 


United States 


6,406 


55,200 


67,700 


129,306 


1,940 


Great Britain 


3,210 


6,100 


46,800 


56,110 


1,470 


Germany 


4,280 


11,500 


29,800 


45,580 


902 


France 


3,380 


9,600 


21,600 


34,580 


910 


Austria 


3,410 


9,900 


9,200 


22,510 


560 


Italy 


2,570 


4,020 


4,800 


11,390 


380 


Spain 


1,540 


5,500 


3,600 


10,640 


590 



5. Of those persons in England and Wales marrying in 1843, 327 
out of every 1000 of the men could not write and 490 out of every 
1000 of the women ; the numbers for each succeeding tenth year to 
1893 were as follows : 304 and 439, 238 and 331, 188 and 254, 126 and 
155, 50 and 57 ; represent these statistics by two curves. 

6. The average annual mortality from smallpox in Sweden has 
been as follows : 

1774-1801, before vaccination, 2045 

1802-1816, vaccination allowed, 480 

1817 to the present, vaccination compulsory, 155 

Represent graphically. 



GRAPHIC ARITHMETIC . 



125 



7. Represent graphically the annexed statistics concerning the 
population of the United States. 

8. The population of the world 
is estimated at 1480 millions dis- 
tributed as follows (in millions) : 
Europe 357.4, Africa 164, Asia 
826, Australia 3.2, the Americas 
121.9, Oceanica and the Polar 
regions 7.5. Represent graphi- 
cally, first arranging in order of 
magnitude. 

9. The distance from the sun 
to the earth is about 93 -10 6 mi., 
to Neptune 30 times as far, to the 
nearest fixed star 256 10 11 ; rep- 
resent these graphically by meas- 
urements on a single straight line. 

10. In a certain city, before the 
strict enforcement of the law re- 
quiring milk and cream to be of a 
certain grade, the following are 
the per cents of samples found 
below grade for the various weeks 
from May 1 to Aug. 1 inclusive : 

50, 64, 42, 45, 42, 35, 35, 38, 37, 39, 45, 37, 28, 50 ; for the corre- 
sponding weeks of the next year, when the law was strictly enforced, 
the per cents are 4, 3, 2, 2, 6, 4, 7, 7, 7, 5, 10, 5, 7, 5. Represent the 
two by broken lines on the same diagram. 

11. The indebtedness of the United States government at the 
various periods named was as follows (in tens of millions of dollars) : 
1791, 7.5; 1800, 8.3; 1810, 5.3; 1816, 12.7; 1820, 9.1; 1830, 4.9; 
1835, 0.0004 ; 1840, 0.5 ; 1850, 6.3 ; 1860, 6.4 ; 1862, 52.4 ; 1863, 112 ; 
1865, 268 ; 1866, 277 ; 1867, 268 ; 1868, 261 ; 1870, 248 ; 1880, 213 ; 
1890, 155; 1896, 179. Represent these statistics graphically, noting 
that the differences in dates are not uniform. 

12. The tonnage of the merchant ships of America and England 
for the various years is here stated in millions ; represent the statistics 
by two curves on the same diagram. America : 1850, 3 ; 1860, 5 ; 
1870, 4 ; 1880, 4 ; 1890, 4.4 ; 1892, 4.8 ; 1894, 4.7 ; 1896, 4.7. Eng- 
land : 1850, 4 ; 1860, 6 ; 1870, 7 ; 1880, 8 ; 1890, 11.6 ; 1892, 12.5 ; 
1894, 13.2 ; 1896, 13.6. 



POPULATION OF THE 
UNITED STATES. 


DECADES. 


TOTAL 
WHITES. 


FOREIGN 
WHITES. 


1750 


1,040,000 




1760 


1,385,000 




1770 


1,850,000 




1780 


2,383,000 




1790 


3,177,257 




1800 


4,306,446 


44,282 


1810 


5,862,073 


96,725 


1820 


7,862,166 


176,825 


1830 


10,537,378 


315,830 


1840 


14,195,805 


859,202 


1850 


19,553,068 


2,244,602 


1860 


26,922,537 


4,138,697 


1870 


33,589,377 


5,507,229 


1880 


43,402,970 


6,679,943 


1890 


54,983,890 


9,249,547 



CHAPTER XIII. 
Introduction to Percentage. 



WITH the introduction of the metric system throughout 
a large part of the world and the almost universal use of 
the decimal system of money save in Great Britain and 
some of her dependencies, the subject of decimal fractions 
has in modern times become one of great importance. It 
has come to be common to reckon by tenths, hundredths, 
and thousandths, and the subject of computation by hun- 
dredths has received the special name percentage. The 
subject requires no principles differing from those used 
in operating with common and decimal fractions, and the 
problems require no methods for solution other than those 
already discussed. There is, therefore, no reason for not 
treating percentage with decimal fractions, as was done to 
some extent, except that it is especially needed in the 
business arithmetic which is now to be considered. 

Common terms. Per cent means hundredths (hundredth , 
of a hundredth), the words, as used in America, always 
being interchangeable within grammatical limits. The 
symbol % means, therefore, either per cent or hundredths 
(hundredth, of a hundredth). 

if TF> 6%, 0.06 are each read 6 per cent, or 6 hundredths. 

!$*% 0.06%, 0.0006 " " 6 hundredths per cent, 6 hundredths 

of a hundredth, 6 hundredths of 
1 per cent, or 6 ten-thousandths. 



PERCENTAGE. 127 

0.01, 1%, Y^Q- are each read 1 per cent or 1 hundredth. 

0.00|, *% " " " * of a hundredth, * per cent, or * of 

1 per cent, and each equals ^. 

The words "per cent" are sometimes taken to mean * out of 100," 
6% then meaning "6 out of 100." 200% would not, however, be so 
clearly understood by this explanation. 

If a certain per cent (meaning a certain number of hun- 
dredths) of a number is to be taken, as 6%, the 6% is 
called a rate, the 6 being called the rate per cent. 

Thus, if the rate of interest in a certain bank is 4%, the rate per 
cent of interest is 4. 

Illustrative Exercises. 
I. 1.55 is what per cent of 15* ? 

1. Let r% = the rate. 

2. Then r% of 15* = 1.55. 

1 fifi 

3. .-. r% = y^ = .10, by dividing equals by 15*. Ax. 7 

II. 69.35 is 9*% of what number ? 

1. Let n = the number. 

2. Then 9*% of n = 69.35. 

3 - ' n = Sir = 730 > bv dividing equals by 9*%. Ax. 7 
o.oy* 

III. What is 10% of $634 ? 

10% of $634 = $63.40. The question is simply, " What is 0.1 of 
$634 ? " No more analysis should be required than in the problem, 
"Find what 2 X $3 equals." 

IV. After deducting 9*% of a number there remains 660.65 ; 
required the number. 

1. Let n = the number. 

2. Then n - 9*% n, or 90*% n, equals 660.65. 

fifift ft^ 

3. .-. n = ^ f- = 730, by dividing equals by 90*%. Ax. 7 



V. After adding 9*% of a number to that number the sum is 
799.35 ; required the number. 

1. Let n = the number. 

2. Then n + 9*%n, or 109*% n, equals 799.35. 

3 - ' n = T^T = 730 ' fe y dividing equals by 109*%. Ax. 7 

'' 



128 HIGHER ARITHMETIC. 

Exercises. 1. The United States silver dollar weighs 26.729 g 
and the Japanese silver dollar (or yen) weighs 26.9564 g ; each is 
0.900 fine (i.e., 90% pure silver); how many grams of fine silver (i.e., 
pure silver) in each ? How do you check your result ? 

2. An English sovereign weighs 7.9881 g and is 0.916 fine ; how 
many grams of fine gold does it contain ? 

3. The British nautical unit of length is the knot, 6080 ft. ; the 
common mile is what per cent of the knot ? 

4. A fathom being strictly 0.1% of a knot, this is what per cent of 
the 6-f t. fathom ? 

5. Of 1486 graduates of women's colleges in England, recently 
questioned, 680 were teachers, 208 were married, 13 were physicians 
or nurses, and the rest were in various professions or trades. What 
per cent of the graduates were teachers ? married ? physicians or 
nurses ? in other work ? How do you check your result ? 

6. The following table shows the values of the total exported mer- 
chandise of the United States for the several years, and of the manu- 
factured part. Find what per cent the manufactured part is of the 
total in each year. 

YEAR. TOTAL. MANUFACTURED. 

1860 $316,242,423 $40,345,892 

1870 455,208,341 68,279,764 

1880 823,946,353 102,856,015 

1890 845,293,828 151,102,376 

1895 793,397,890 183,595,743 

7. From the data of Ex. 6 find the rate of increase of each amount 
over that of the preceding period. 

8. The ratio of the arid and semi-arid regions of the United States 
(excluding Alaska) to the rest of the country is about 24 : 25 ; at 
this ratio, what per cent of our territory is arid and what per cent is 
semi-arid ? 

9. From the following table showing the wealth of the United 
States and the average wealth of each inhabitant, compute the rate 
of increase in each from period to period. 

CENSUS. MILLIONS OF DOLLARS. DOLLARS PER INHABITANT. 
1820 1,960 205 

1840 3,910 230 

1860 16,160 514 

1880 43,642 870 

1890 65,037 1,039 



PERCENTAGE. 129 

10. Cinnabar consists of two substances, sulphur and mercury, in 
the ratio of 7 parts (by weight) of the former to 44 of the latter. The 
weight of the sulphur is what per cent of the weight of the mercury ? 
That of the mercury is what per cent of that of the sulphur ? The 
weight of each is what per cent of the weight of the cinnabar ? How 
many grams of each in 178.5 g of cinnabar ? 

11. A dealer is obliged to sell sugar so that for 43.5 Ibs. he receives 
as much as 36 Ibs. cost ; did he gain or lose, and what rate per 
cent? 

12. At the time of a recent census in Ireland 38,121 people, or 
0.81% of the total population, could speak the Irish language only ; 
required the population at that time, correct to 1000. 

13. In 1894 the population of London was 4,349,116, an increase 
of about 3.28% over the population in 1891 ; required the population 
in 1891, correct to 1000. 

14. National banks are required to keep on hand 25% of their 
deposits ; find if these banks have complied with the law and give the 

per cent in each case : 

SPECIE ON OTHER LEGAL ^ 
HAND. TENDER. ^POSITS. 

(a) BankofN.Y. $2,050,000 $1,200,000 $12,170,000 

(6) Manhattan Bank 2,608,000 3,219,000 16,154,000 

(c) Nassau Bank 192,000 539,400 2,885,000 

(d) German Exchange Bank 267,900 587,400 3,140,500 

(e) Germania Bank 511,400 541,400 4,196,300 

15. A certain bank has on hand $294,600 in specie and $325,400 
in other legal tender, and this sum is 26.4% of the deposits ; find, 
correct to $1000, the amount of the deposits. 

16. A report made by the banks of New York City shows 
$164,172,200 in cash on hand, this being 31.3% of the total deposits ; 
find, correct to $1000, the amount of the deposits. 

17. In one year the imports of specie into New York amounted to 
$83,233,962, and the exports to $102,487,994 ; the difference was 
what per cent of the imports ? of the exports ? 

18. An insurance company charges a premium of $22.50 for insur- 
ing a house for $1500 for 3 yrs. ; what is the rate for the 3 yrs.? 

19. A book agent sells during the summer 300 books at $2.75 each ; 
he is allowed 40% of the receipts ; how much does he earn ? 

20. A man invests $3000 in property which he rents for $228 a 
year. The taxes are $33, insurance is $18, water tax $5, repairs $47; 
what per cent does he receive on his investment ? 



130 HIGHER ARITHMETIC. 

21. What is the per cent of attendance in a schoolroom of 59 
pupils when there are 29-J- da. of absence in 4 school weeks ? when 
there are 10 da. of absence in 1 school week ? 

22. A man sold two horses for $125 each ; on the purchase price 
of one he gained 20% and on that of the other he lost 20% ; what was 
his total gain or loss ? 

23. The distances between the following cities by the present 
routes of sea travel and also by the proposed Nicaragua Canal are 
given below ; required the gain per cent by the canal over the present 
route, in each case. 

MILES VIA 

BETWEEN MILES, PRESENT NICARAGUA 

ROUTE, VIA CANAL 

(a) New York and San Francisco Cape Horn 15,660 4,907 

(6) New York and Puget Sound Magellan 13,935 5,665 

(c) New York and Hong Kong Cape G. H. 13,750 10,695 

(d) New York and Melbourne Cape Horn 13,760 9,882 

(e) Liverpool and San Francisco Cape Horn 15,620 7,627 
(/) New Orleans and San Francisco Cape Horn 16,000 4,147 

24. A man has the following investments : $2000 which yields 4%, 
$450 which yields 6%, and $1200 which yields 6|%. He can invest 
the whole amount so that it will yield 5% ; would he gain or lose by 
so doing, and what per cent of the whole sum ? 

25. What is the premium for insuring a house for $3000 for 3 yrs. 
at l-J-% for the whole time ? at 2% ? at 75 cts. per $100 ? at $7 per 
$1000 ? 

26. It is estimated that about 600,000,000 passengers are carried 
on steamboats in the United States in one year and that about 70 are 
killed ; what per cent are not killed ? 

27. About 590,000,000 passengers are carried on railways in the 
United States in one year and about 300 are killed ; what per cent 
are not killed ? 

28. About 75f % of the number of patents granted by the United 
States in the 60 yrs. beginning with 1837 represents the number of 
patents refused ; the total number of applications was 993,953 ; how 
many patents were granted and how many refused ? (Answer correct 
to 1000.) 

29. By the eleventh census the number of Indians on reservations 
under control of the Indian Office was 106% as large as the rest of the 
Indians ; the total number was 249,000 ; how many were in each of 
these two groups ? (Answer correct to 1000.) 



PERCENTAGE. 131 

30. About 74% of the territory of the United States, excluding 
Alaska and the Indian reservations, is cleared land, and 495,000,000 
acres are forest; what is the total area, excluding the portions specified? 

31. What per cent of the 343,267 immigrants landing in the United 
States in a certain year did not rank among either the 46,807 skilled 
laborers or the 2324 professional men ? 

32. The chief export of the United States is unmanufactured 
cotton, which is worth about 8 cts. a pound, and the value of which 
is 22% of the total exports of merchandise which amounted in a certain 
year to $863,200,487 ; how many pounds of cotton were exported ? 
(Answer correct to 1,000,000.) 

33. If the total value of merchandise imported into the United States 
in a certain year was $731,969,965, 13% being coffee of which there 
were 652,000,000 Ibs. ; what was the average price per pound of coffee ? 

34. The cost of collecting the customs revenue of $160,021,752 in 
a certain year being 4.52%, and of collecting the internal revenue of 
$146,762,865 being 2.62%, the former netted the government how 
much more than the latter ? 

35. The world's total production of wool in a certain year being 
2,582,103,000 Ibs., and the four largest producers being Russia with 
290,000,000 Ibs., Argentina with 280,000,000 Ibs., the United States 
with 272,475,000 Ibs., and Great Britain with 135,000,000 Ibs., find 
what per cent of the total these countries produced, both collectively 
and separately. (Answer correct to 0.1%.) 

36. The following table shows the receipts and certain disburse- 
ments of "old line" life assurance companies reporting to the N. Y. 
Insurance Department : 

1,00MB. POUd. s 

1871 $113,490,562 $28,773,041 $14,624,608 $20,242,707 
1895 266,897,200 84,791,622 15,297,604 62,052,872 

Find the rate of increase in each column. 

37. Ascertain the population of the city or village in which you 
reside, according to the last three census reports ; represent the statis- 
tics graphically and compute the rate of increase or decrease of popu- 
lation for each period. 

38. Similarly for the average annual attendance of your school for 
the past five years. 

39. The radius of the sun being 10,856% of that of the earth, the 
latter being 6370 km, compute the volume of the sun in cubic kilo- 
meters. 



CHAPTER XIV. 
Commercial Discounts and Profits. 



IT is the custom of manufacturers, publishers, and whole- 
sale dealers to fix a price for their products and then to 
allow a discount under certain conditions. 

E.g., a book may be published at $2.00 with a discount of 25% to 
dealers, the book costing them $2.00 25% of $2.00, or $1.50. The 
$2.00 is known as the list price, the $1.50 as the net price. 

It frequently happens that wholesale houses issue expen- 
sive catalogues in which prices are specified. But as the 
cost of production varies, these prices change, and in order 
not to issue a new catalogue a house will print a new list 
of discounts for its customers. In some lines, indeed, the 
catalogue price has been so long fixed as to be several hun- 
dred per cent above the actual price, the latter being fixed 
by the discounts, of which there are often several. 

E.g., paper bags are quoted at a certain price, but the bill sent to 
the retailer may read "Less 70% 25% 10%, 30 da., and 2% off in 10 da." 
This means that they can be produced so much cheaper than formerly 
that the purchaser is allowed a discount of 10%, then 25% from that 
price, then 70% from that, and finally, if he pays within 10 days instead 
of waiting 30 days, he is allowed a further discount of 2%. Hence, if 
the list price was $100, the net price would be 

$90 after deducting 10% of $100.00, 
67.50 " " 25% " 90.00, 

20.25 " " 70% " 67.50, 

19.84 " " 2% " 20.25. 



COMMERCIAL DISCOUNTS. 133 

In this case, the catalogue price is over 500% of the actual price paid. 
In Ex. 6 it is shown that it is immaterial in what order these discounts 
are taken. 

On the bill heads of wholesale houses there is usually a note show- 
ing what discounts, if any, are allowed. For example, "Terms: 30 
da. net, 1% 10 da." ; " Terms : 60 da., or 2% if paid within 10 da." ; 
" Terms : Net 60 da., or 2% disct. if paid in 10 da." 

Illustrative problems. 

1. The list price of some goods is $62.70, a discount of 10% 6% 3% 
being allowed ; required the net price. 

Solution. 0.97 0.94 0.90 of $62.70 - $51.45. 
Analysis. 1. Let I = list price, $62.70. 

2. Then I 0.10 1 = 0.90 J, the remainder after the first discount. 

3. Then 0.90 1 0.06 0.90 1 = 0.94 0.90 1 = the remainder after the 
second discount. 

4. Similarly, 0.97 -0.94 0.90 1 = the remainder after the third dis- 
count = net price. 

Application of logarithms. If the student has studied Chap. XI, 
this furnishes an application, the answer requiring no more than four 
figures and thus coming within the range of the table on pp. 114, 115. 

log 0.97 = 0.9868 - 1 

log 0.94 = 0.9731 1 

log 0.90 = 0.9542 - 1 

log 62.70 = 1.7973 

log 61.45 = 1.7114 

Unless logarithms are used, which is not advisable in practice, it 
is, of course, better to take 10% of $62.70 and subtract, then 6% of 
this remainder and subtract, and then 3% of this remainder and sub- 
tract, than to perform the multiplication by 0.97-0.94-0.90. 

II. A merchant sells goods at a discount of 25% from the marked 
price and still makes a profit of 25% on the cost ; at what per cent 
above cost did he mark them ? 

1. Let c = the cost, and m = the marked price. 

2. Then m 0.25 m = c + 0.25 c. 

3. .-. 0.75 m = 1.25 c. 






5. .-. he must mark them 66% above cost. 



134 HIGHER ARITHMETIC. 

Exercises. 1. The list prices and rates of discount being as fol- 
lows, find the cost : 

LIST PRICE. RATES OF DISCOUNT. 

(a) $1271.50 33%, often called " a third off." 

(6) 3.00 25%, " " "a quarter off." 

(c) 125.00 15%. 

(d) 37.50 20% 12i% 6%. 

(e) 2107.50 30% 8%. 

(/) 403.80 25% 10% 4%. 

(g) 3462.95 10% 3%. 

(h) 178.65 12|% 8% 2%. 

(i) 83.90 15% 7% 3%. 

(j) 623.30 8% 2% 1%. 

(k) 375.00 25% 10% 6%. 

(/) 150.00 a third off 40%. 

2. In each case of Ex. 1, suppose the buyer had sold the articles at 
the list price, what would have been his rate of gain on the cost ? 

3. In Ex. 1, what one rate of discount would have been equivalent 
to the several rates mentioned in (d), (e), (I) ? 

4. Suppose a dealer buys goods at " a third off " and sells them at 
" a quarter off " the list price, what is his rate of gain on the cost ? 

5. Show that the discounts 10% 8% 3% are equivalent to the dis- 
counts 3% 8% 10%, but not to the single discount 10% + 8% + 3%. 

6. Generalizing Ex. 5, show that the discounts r\% r 2 % r s % are 
equivalent to the discounts r 8 % r 2 % ri%, or r 2 % r 3 % ri%, etc. ; that is, 
that it is immaterial in what order the discounts are taken. 

7. The cost of certain goods and the rates of discount being as fol- 
lows, find the list prices : 

COST. RATES OF DISCOUNT. 
(a) $1827.40 12*%. 

(6) 436.90 25% 10%. 

(c) 49.63 30% 12% 6%. 

(d) 2341.50 30% 10% 2%. 

(e) 693.49 25% 10% 3%. 
(/) 127.90 33*% 6%. 
(g) 647.00 20% 8% 1%. 

8. A merchant buys goods listed at $250, on which a discount of 
15% 10% 3% is allowed ; he sells the goods for $225 ; what rate of 
profit does he make on the cost ? 



COMMERCIAL DISCOUNTS. 135 

9. Prove that if the rates of discount are ri, r 2 , the equivalent 
single rate of discount is r\ + r 2 r\- r 2 , and hence that the single 
rate equivalent to two rates of discount equals their sum minus their 
product. 

10. A bill of merchandise amounting to $327.50 was bought Oct. 1, 
"Terms: 3 mo. or 5% off 60 da., or 10% off 30 da." How much 
money would settle the bill Jan. 1 ? Nov. 20 ? Oct. 27 ? 

11. What is the cost of a bill of hardware amounting to $1027.40, 
discounts 40% 10% 3%, freight being $10.60 ? 

12. What is the net value of one case of prints containing 3000 
yds. @ 6 cts. per yd., less 8% discount ; package $0.40, freight $0.95 ? 

13. A merchant buys goods at a discount of 30% 20% and sells them 
at % off the list price ; what is his gain per cent on the cost ? 

14. Which is the better for the buyer to take, a discount of 

(a) 30% 15% 10% or one of 47% ? 

(6) 10% 10% 5% 23%? 

(c) 15%12|% " 25%? 

(d) 12% 8% 1% 19%? 

(e) 10% 6% 2% 17%? 
(/) 30% 30% 30% " 66%? 

15. At what per cent above cost must goods be marked in order to 
take off from the marked price 

(a) 10% and still make a profit of 8% on the cost ? 

(6) 25% " " 20% 

(c) 20% " " 30% " 

(d) 10% " " 50% 

(e) T!% " " r 2 % 
(/) 15% and lose 5% 
(9) 20% 15% 
(ft) n% " r 2 % 
(i) 30% and neither gain nor lose ? 

16. A dealer purchases some goods listed at $281.50, off and 5% 
for cash ; if he pays on delivery, what is the net price, and what per 
cent discount can he give on the list price in order to make a profit of 
15%? 

17. Three rates of discount, 10% 10% r% are equivalent to the 
single rate 27.1% ; find r. 

18. What are the three equal rates of discount equivalent to the 
single rate 48.8% ? 



CHAPTER XV. 
Interest, Promissory Notes, Partial Payments. 



I. SIMPLE INTEREST. 

THERE is practically a single type of problem in this 
subject, given the principal, rate, and time, to find the 
interest. Bankers and others who frequently meet this 
problem find the interest by the help of printed tables. 
These tables are usually based on 360 days to the year, 
and in using them it is customary to reckon exact days 
between dates. Some banks use tables based on 365 days 
to the year, this being the fairer method although yielding 
less interest. 

People generally, working without tables, reckon 360 
days to the year, but they find the difference between 
dates by subtracting months and days, calling 30 days 
1 month, and this is the method used in this work. 

Thus, to find the time from July 2 to Sept. 2, it is customary, if 
one has no interest table, to say 

9 mo. 2 da. 
7 " 2 " 
2 mo. 

But if an interest table is at hand, it will readily appear that 

Sept. 2 is the 245th day of the year, 
and July 2 " 183d " " 
and that the difference in time is 62 da. 



SIMPLE INTEREST. 



137 



For exercises in analysis certain other problems are 
usually given in school, though rarely met in business. A 
few such have been inserted, types appearing in Problems 
II, III, IV, VI, on p. 138. 

Interest is reckoned as a certain per cent of the prin- 
cipal. When a rate is specified, the words " for one year " 
are to be understood unless the contrary is stated. 

7.e., if it is said that the rate of interest is 6%, it means that the 
interest for 1 yr. is 6% of the principal. Occasionally, however, 
interest is quoted by the month, as 1% a month. 

Interest table. The following represents the first part 
of a page from an interest table such as bankers use. 

This particular page, if printed in full, would give the interest at 
6% for 3 mo. and any number of additional days from to 29. This 
portion gives only 0, 1, 2, 3 da. in excess of 3 mo., this being sufficient 
for illustration. 



3 MONTHS. 



TOTAL 
DAY*. 


1000 


2000 


3000 


4000 


5000 


6000 


7000 


8000 


9000 


DAYS OVER 
3 Mo. 


90 


15.00 


30.00 


45.00 


60.00 


75.00 


90.00 


105.00 


120.00 


135.00 





91 


15.17 


30.33 


45.50 


60.67 


75.83 


91.00 


106.17 


121.33 


136.50 


1 


92 


15.33 


30.67 


46.00 


61.33 


76.67 


92.00 


107.33 


122.67 


138.00 


2 


93 


15.50 


31.00 


46.50 


62.00 


7750 


93.00 


108.50 


124.00 


139.50 


3 



The method of using the table will be seen from the following 
computation of the interest on $3975 for 93 da. at 6% : 
Int. on $3000 = $46.50 
900 = 13.95, T V of int. on $9000. 
" " 70= 1.09 
" 5= .08 

$61.62 

It will be noticed that the interest on ordinary sums can be told by 
merely glancing at the table. Thus, the interest on $250 for 3 mo. is 
$3.00 + $0.75 = $3. 75. 



138 HIGHER ARITHMETIC. 

Illustrative Problems. 

I. What is the interest on $360 for 1 yr. 6 mo. 10 da. at 6% ? 

1. 1 yr. 6 mo. 10 da. = lf yrs. 

2. Int. for 1 yr. = 6% of $360. 

W 10 

3. .-. int. for lf yrs. = 0.06 $200 = $33. 

pp 

II. The interest on $360 for 1 yr. 6 mo. 10 da. is $33 ; required 
the rate. 

1. 1 yr. 6 mo. 10 da. = lf yrs. = ff yr. 

2. v the int. for ff yr. = $33, 

... i* u lyr . =$33 -5.^ = 1$. |33. 

3. vr%of$360 = f-$33, 

3 



5 10 

III. How long will it take the interest on $360 at 6% to equal $33 ? 

1. The int. for 1 yr. = 6% of $360. 

2. .-. " " tjrs.=t- 6% of $360 = $33. 

$33 

0.06 -$360 3ir ' 
4. .-. it will take lf yrs., or 1 yr. 6 mo. 10 da. 

IV. On what sum of money will the interest for !$ yrs. at 6% 
equal $33 ? 

1. Let p = the principal. 

2. v the int. for If! yrs. at 6% on p is $33, 

3. .-. 

V. Find the amount (principal plus interest) of $360 for 1 yr. 
6 mo. 10 da. at 6%. 

1. By Prob. I, amt. = $360 + f f 0.06 $360 

= (1 + f f . 0.06) $360 = $393. 

VI. What principal will amount to $393 in |f yr. at 6% ? 

1. Rateforffyr. =ff-0.06. 

2. Let p the principal ; then, 

p + ff -0.06 p = $393, or 
(1 + |f -0.06) p = $393. 




SIMPLE INTEREST. 139 

Exercises. 1. Find the interest on $10,000 from July 2 to 
Sept. 2, at 6%, reckoning the time as follows : 

(a) Subtract the months and days, calling 30 da. = 1 mo., and 
360 da. = 1 yr. 

(&) Take exact days between dates, but let 360 da. = 1 yr. 

(c) Subtract as in (a), but let 365 da. = 1 yr. 

(d) Take exact days between dates, but let 365 da. = 1 yr. 

2. Of the four plans given in Ex. 1, 
(a) Which gives the most interest ? 

(6) Which is easiest without interest tables ? 

(c) Which is the fairest ? 

(d) Which result differs most from the fairest result ? 

(e) Why do people generally, without interest tables, use method (a) ? 
(/) Why do bankers generally use method (6) ? 

(g) Why is method (c) not used ? 

(h) Why do the government and some banks use method (d) ? 

3. Find the interests on the following principals for the times and 
at the rates specified : 

(a) $250 for 2 yrs. 4 mo. 8 da. at 6%. 

(6) $40 " 1 " 3 " " 5%. 

(c) $125 " 8 " 15 " " 7%. 

(d) $350 " 3 " 6 " " 4%. 

(e) $820 " 6 " 4i%. 
(/) p " t " " r%. 

4. Find the rates at which the following principals will yield the 
interests mentioned in the respective times : 

(a) $300 yields $45 interest in 2 yrs. 6 mo. 
(6) $175 " $10.50 " 1 " 6 " 
(c) p " i " t " 

5. Find the times in which the following principals will yield the 
interests mentioned at the respective rates ; 

(a) $450 yields $24 at 6%. 
(6) $125 $6.25 " 4%. 
(c) p i r%. 

6. Find the principals which will yield the following interests at 
the times and rates mentioned : 

(a) $62.50 interest in 1 yr. 3 mo. at 4%. 

(6) $5 " 1 " 1 10 da. * 6%. 

(c) i " t " " r%. 



140 HIGHER ARITHMETIC. 

7. Find the principals which will amount to the following sums at 
the times and rates specified : 

(a) $280 in 2 yrs. at 6%. 

(6) $45.25 " 1 " 10 mo. 15 da. " 7%. 
(c) a " t " " r%. 

8. From the portion of the interest table given on p. 137, find the 
interest at 6% on : 

(a) $275 for 3 mo. 

(6) $750 " 3 " 3 da. 

(c) $9275 " 91 " 

(d) $5750' " 93 " 

Short methods. Before interest tables were common, 
short methods of computing interest were valuable. At 
present those who have much work of this kind use these 
tables. One method is, however, of enough value to be 
mentioned, especially as the most common rate of interest 
is 6%, and as most notes run for 90 days or less. 

Required the interest on $250 for 63 da. at 6%. 

1. v the rate for 1 yr. is 6%, 

2. .-. " $ " , or 2 mo., is 1%. 

3. 1% of $250 = $2.50, interest for 60 da. 

4. ^ "$2.50 = $0.12i, " 3 " 

5. .-. $2.62 = " 63 " 

In practice it is merely necessary to put down these figures, the 
vertical line representing the decimal point : 

$2150 60 da. 

|l2j 3 " 

$2|62i 63 da. 

For 7%, add \ of $2.62^, and similarly for other rates. 

Exercises. Find the interests on the following sums for the times 
and at the rates specified : 

1. $144, 30 da., 6%. 

2. $750, 93 " 6%. 

3. $125, 60 " 6%. 

4. $250, 93 6%. 

5. $400, 33 " 7%. 

6. $50, 90 " 5%. 

7. $150, 60 '" 8%. 



PROMISSORY NOTES. 141 



II. PROMISSORY NOTES. 

Most promissory notes between individuals are of sub- 
stantially the following form : 

f 500. Chicago, 111., Dec. 3, 1900. 

Thirty days after date, I promise to pay to John Jones, 
or order, five hundred dollars, value received, with interest 
at 5%. John Smith. 

1. In this case John Smith is the maker, John Jones the payee, 
$500 the face, and the face plus the interest is the amount, or future 
worth. 

2. This note is negotiable, and may be sold by the payee, the trans- 
fer being indicated by his indorsing the note, that is, by writing his 
name across the back. Notes payable to the payee "or bearer" are 
also negotiable. 

3. By indorsing the note the payee becomes responsible for its pay- 
ment in case the maker does not pay it. But if the buyer is willing 
to take the note without this guarantee, the indorser may be released 
by first writing the words " Without recourse" across the back, and 
then his name. 

4. The indorsement may be made in blank, that is, the payee may 
merely write his name across the back, or in full, that is, the payee 
may specify the person to whose order it is to be paid, thus : 

"Pay to the order of John Brown. 

John Jones." 

5. A note matures on the day when it is legally due. When the 
time is specified in days, exact days are counted in ascertaining 
maturity ; when in months, calendar months are counted. 

6. Many states, following an old custom, allow three days of grace 
for the payment of notes. That is, a note dated Dec. 3, the time being 
" 30 days after date," is legally due Jan. 2 + 3 days, or Jan. 5, a fact 
indicated by writing "Due Jan. 2/5." A considerable number of 
states have abolished these days of grace, and the custom will in time 
become obsolete. Where the law still allows them it is quite common 
for notes to bear the words " Without grace." 

7. The law as to the time of payment of notes due on legal holidays 
varies in different states. 



142 HIGHER ARITHMETIC. 

8. If a note reads " with interest," but does not specify the rate, 
it draws the rate specified by the law of the state. If it does not call 
for interest, it draws none until it is due and payment is demanded, 
after which it draws the legal rate. 

9. In some states the law specifies what is called the " legal rate," 
and then specifies a maximum rate above which no contract for interest 
is legal. In other states the u legal rate " is also the maximum. Some 
states specify no maximum, allowing the parties to the contract to fix 
any rate they wish. Interest above the maximum rate allowed by law 
is called usury, and the taking of usury is punished according to the 
laws of the various states in which it is forbidden. 

Notes payable at a bank are discussed in Chap. XVI on 
Banking Business. The protest of notes is also discussed 
in that connection. 

Exercises. 1. In your state, are days of grace allowed on prom- 
issory notes ? When are notes which mature on legal holidays pay- 
able in your state ? 

2. What is the " legal rate " of interest in your state ? Is there a 
maximum rate beyond this ? What is the punishment for usury in 
your state ? 

3. What is the rate at which money is usually loaned to responsible 
persons in your vicinity ? 

4. Write a 90-day note, signed by Peter Brown and payable to 
your order, bearing the rate which you found in Ex. 3 ; indorse it so 
that it shall be payable to the order of Kobert Jones. 

5. Find the dates of maturity of, the interests on, and the amounts 
of promissory notes for the following sums, at the specified rates, 
supposing the notes paid when due ; add the days of grace if such is 
the law in your state, otherwise not : 

(a) $500, dated Feb. 7, due 6 mo. after date, at 6%. 

(6) $250, " Mar. 1, " 1 yr. 6 mo. " 5%. 

(c) $100, " July 15, " 90 da. " 7%. 

(d) $750, " Sept. 7, " 2 yrs. " 4|%. 

(e) $1275, " Aug. 10, " 60 da. " 6%. 

(/) $350, " June 3, " 4 mo. " at the rate found 

in Ex. 3. 

(g) $50, " Dec. 10, " 2 " " " 

(h) $200, " Oct. 5, " 4 " " 



PARTIAL PAYMENTS. 143 



III. PARTIAL PAYMENTS. 

If a note or other obligation draws simple interest, and 
partial payments have been made at various times, the sum 
due at any specified date is usually computed as follows : 

1. The interest on the principal is found to that time 
when the payment or payments which have been made 
equal or exceed this interest. 

2. The payment or payments are then deducted and the 
remainder is treated as a new principal. 

These directions constitute what is known as the United States Rule 
of Partial Payments, the only legal method in most states. A few 
states, however, require other methods, and in these the teacher 
should explain the law and require the problems solved accordingly. 

The United States Rule, and the reason for the first sentence, will 
be understood from a single problem : A note for $1000 is dated Jan. 2, 
1900, and draws 6% interest ; the following payments have been made, 
Jan. 2, 1901, $1 ; July 2, 1901, $89; Sept. 2, 1901, $500; required 
the amount due Jan. 2, 1902. 

1. On Jan. 2, 1901, the $1000 amounts to $1060. 

2. If the $1 were now deducted the new principal would be $1059. 

3. But then the borrower would be paying interest on $59 more 
than he agreed. 

4. .-. it would not be right to deduct the $1, or any other sum 
which might be paid, unless it should equal at least $60. 

5. The practical solution usually appears in the following form : 



1901 
1900 


7 mo. 
1 " 


2 da. 
2 " 


Int. to July 2, 
Amt. 
Paymts. " 
New prin. " 
Int. to Sept. 2 
Amt. " 
Paymt. " 
New prin. " 
Int. to Jan. 2, 
Amt. " 


1901, 

($1 + $89), 
, 1901, 

1902, 


$1000. 
90. 


1 

1902 
1901 


6 

9 

7 


2 

2 


$1090. 
90. 


$1000. 
10. 


2 

1 

9 


2 
2 


$1010. 
500. 


$510. 
10.20 




4 




$520.20 



144 



HIGHER ARITHMETIC. 



Payments less than the accrued interest are seldom made. When 
they are made it is usually possible to detect the fact that they are 
less than the interest, before computing the amount due on that 
date. 

Partial payments are usually indorsed on the note, that is, written 
across the back, as, for example : 

"Jan. 2, 1902. Rec'd $10. 
July 5, 1902. Rec'd $50." 

Exercises. Find the amounts due at the dates of settlement 
specified : 



DATE OF NOTE. 


FACE. 


BATE. 


PARTIAL PAYMENTS. 


SETTLED. 


1. Jan. 10 


$603 


6% 


June 1, $100; Aug. 15, $50; Sept. 


Oct. 15 








20, $30 




2. Apr. 4, 1900 


$125 


7% 


Aug. 1, $5; Oct. 16, $30; Jan. 10, 


Apr. 1, 1901 








1901, $75 




3. Nov. 1, 1899 


$50 


6% 


Dec. 12, $5; Jan. 10, 1900, $40; 


Apr. 10, 1901 








Feb. 1, $1 




4. Jan. 11, 1899 


$500 


5% 


Jan. 11, 1900, $20; July 11, $15; 


Mar. 5, 1901 








Jan. 11, 1901, $50 




5 June 14, 1897 


$375 


6% 


Sept. 10, $3.50; Nov. 15, $4.75; 


June 21, 1899 








Jan. 7, 1898, $51.75; Jan. 11, 










1899, $200 




6. May 1, 1897 


$1000 


5% 


Sept. 1, $5; Nov. 1, $3; Mar. 1, 


Sept. 1,1898 








1898, $100; July 15, $275 




7. Apr. 1, 1901 


$200 


4i% 


July 1, $50; Jan. 16, 1902, $5; 


Sept. 1,1902 








June 10, 1902, $2; July 1, 1902, 










$75 




8. June 1, 1898 


$800 


5% 


Jan. 3, 1899, $35; Aug. 1, $15; 


Dec. 1,1902 








Nov. 3, 1899, $70; Aug. 16, 1900, 










$100; Feb. 1, 1901,'$125; Sept. 










1,1902, $180 




9. Apr. 1, 1901 


$500 


6% 


Jan. 1, 1902, $100; Aug. 7, 1903, 


Jan. 1,1904 








$25 




10. Jan. 1, 1900 


$2000 


6% 


Jan. 1, 1901, $500; Apr. 1, 1902, 


Nov. 13, 1905 








$250; Dec. 16, 1903, $100; Jan. 










1, 1905, $600 




11. Feb. 5, 1904 


$675 


5% 


Apr. 1, 1905, $25; Aug. 5, 1905, 


Jan. 20, 1907 








$100; Sept. 5, 1905, $50; Jan. 










20, 1906, $200 




12. May 2, 1900 


$575 


5% 


July 1, 1901, $75; Sept. 3, 1901, 


Sept. 17, 1904 








$100; Jan. 1, 1902, $50; Apr. 1, 










1902, $100; July 1, 1902, $100; 










Sept. 17, 1903, $50 





COMPOUND INTEREST. 145 



IV. COMPOUND INTEREST. 

Savings banks usually add the interest to the principal 
at the end of the interest period, say every six months. 
The whole amount then draws interest, the depositor thus 
receiving interest on interest, or compound interest. Other- 
wise, the subject is not often met in a practical way, 
although banks, by loaning their interest as it is received, 
really have all of the benefits of compound interest. 

As in simple interest, there is a single case of practical 
value given the principal, rate, and time, to find the com- 
pound interest or the amount. 

E.g., what is the amount of $500 for 3 yrs. at 3% compound 
interest ? 

1. The amt. of $500.00 and int. for 1 yr. = $515.00. 

2. " $515.00 " " =$530.45. 

3. " $530.45 " " =$546.36. 

4. .-. " $500.00 " 3 yrs. = $546.36. 

Similarly, what is the amount of $150 for 3 yrs. at 4%, interest 
compounded semi-annually ? 

1. The amt. of $150 and int. for 6 mo.= $150 + 0.02 $150 

= 1.02 -$150. 

2. .. " 1.02 -$150 " = 1.02 -1.02 -$150 

= 1.02 2 -$150. 

3. .-. " 1.02 2 -$150 " = 1.02* $150. 

4. And finally, the amount for 6 six-month periods 

= 1.026 -$150 = $168.93. 

Interest tables. While compound interest is not in 
general use, it frequently happens that large investors 
wish to compute the amount resulting from reinvesting all 
interest as it becomes due; in other words, they wish to 
ascertain the amount of a certain sum at compound interest. 
For this purpose they resort to compound-interest tables, 
a specimen of which is given on p. 146. A table of loga- 
rithms evidently answers the same purpose. 



146 



HIGHER ARITHMETIC. 



AMOUNT OF $1000 AT COMPOUND INTEBEST. 



YEABS. 


2% 


2% 


3% 


4% 


5% 


6% 


1 


1020.00 


1025.00 


1030.00 


1040.00 


1050.00 


1060.00 


2 


1040.40 


1050.63 


1060.90 


1081.60 


1102.50 


1123.60 


3 


1061.21 


1076.89 


1092.73 


1124.86 


1157.63 


1191.02 


4 


1082.43 


1103.81 


1125.51 


1169.86 


1215.51 


1262.48 


5 


1104.08 


1131.41 


1159.27 


1216.65 


1276.28 


1338.23 


6 


1126.16 


1159.69 


1194.05 


1265.32 


1340.10 


1418.52 



If the interest is at the rate of 4%, 5%, or 6% per year, but com- 
pounded semi-annually, the amount is evidently the same as if the rate 
were 2%, 2-J-%, or 3%, respectively, compounded annually for a period 
twice as long. 

jEJ.gr., what is the amount of $2750 for 3 yrs. at 5%, compounded 
semi-annually ? 

Amt. of $1000 for 6 yrs. at 2|% compounded annually = $1159.69. 
" $2750 = 2.75 X $1159.69 = $3189.14. 

The subject of compound interest is still further dis- 
cussed in the Appendix, Note III. 

Exercises. Find the amounts of the following sums for the times 
and rates of compound interest specified : 

1. $50, 2 yrs. 6 mo., 3%, compounded semi-annually. 

" annually. 

u semi-annually. 

" quarterly. 



4%, 



$168, 4 yrs. 3 mo., 

$1200, 3 yrs. 2 mo., 

$350, 1 yr. 8 mo., 4%, " 

$p, 1 yr., r%, " annually; also for 

2 yrs., 3 yrs., t yrs. 

6. From Ex. 5, find p, the principal, which at r%, compounded 
annually for t yrs. , amounts to a. 

7. From Ex. 6, find p, given a = $123.73, t - 3, r% = 4%. 

8. From the compound-interest table, find the amount of $500 at 
4%, compounded annually for 5 yrs. 

9. Also of $2500 at 5%, compounded semi-annually for 2 yrs. 

10. Also of $350 at 3%, compounded annually for 6 yrs. 

11. Also of $4000 at 2%, compounded annually for 4 yrs. 



ANNUAL INTEREST. 147 

V. ANNUAL INTEREST. 

In some states, if a note or bond contains the words 
"with interest payable annually" this interest, if left 
unpaid, also draws interest to the day of settlement, or 
until cancelled by payment. The note or bond is then 
said to draw annual interest. 

E.g., to find the amount due on a $500 note dated Jan. 1, 1900, 
drawing annual interest at 6%, no payments made until the day of 
settlement, Jan. 1, 1904. 

1. Face of note = $500. 

2. Int. on $500 for 4 yrs. at 6% = 120. 

3. Int. on $30 for 3 yrs. + 2 yrs. + 1 yr. at 6% = 10.80 

4. Amt. due Jan. 1, 1904 = $630.80 

Unless annual interest is allowed in the state in which this book is 
used, this subject may be omitted. 

Semi-annual or quarterly interest is treated hi a similar manner. 

Exercises. 1. What is the amount due at the end of 3 yrs. on a 
$1000 note bearing annual interest at 5%, no payments having been 
made? 

2. In the western states coupon notes are often given, that is, notes 
bearing coupons which are themselves promissory notes for the interest 
due, and also drawing interest, often at a higher rate. Find the 
amount of a coupon note for $1000 at the end of 5 yrs., the principal 
drawing 6%, the coupons representing the interest due annually and 
drawing 8% remaining unpaid. 

3. A coupon note draws 6%, the coupons being due semi-annually 
and drawing 10% if unpaid ; the face of the note being $800, and no 
payments having been made, find the amount due at the end of 4 yrs. 

4. What is the amount due at the end of 4 yrs. on a $300 note 
bearing 5% interest, payable semi-annually, no payments having been 
made ? 

5. A coupon note draws 6%, the coupons being due semi-annually 
and drawing 8%, if unpaid ; the face of the note being $500, and no 
payments having been made, find the amount due at the end of 3 yrs. 

6. In Ex. 5, supposing the first three coupons had been paid when 
due, find the amount due at the end of 3 yrs. 



CHAPTER XVI. 
Banking Business. 



THE ordinary business of a bank is largely included 
under the following heads : 

1. Receiving deposits and paying from the same on 
presentation of checks signed by the depositor. 

2. Lending money upon promissory notes or (chiefly in 
the case of savings banks) upon bonds and mortgages. 

3. Discounting notes which individuals may own and 
upon which they wish to realize money before the notes 
are due. 

4. Selling drafts on other banks, and collecting drafts 
drawn by one person or corporation on another. (See 
Chap. XVII.) 

I. DEPOSITS AND CHECKS. 

If a person deposits money in a savings bank (or in the 
savings department of a bank having both savings and 
commercial departments), he usually receives a book in 
which are written the sums deposited and drawn out. If 
he wishes to draw out any money, he presents his book for 
the debiting of the amount and is usually required to sign 
a receipt. Savings banks usually pay from 2% to 4% 
interest compounded semi-annually. 

Ordinary deposits in other banks do not draw interest, 
the deposit being made for convenience and safety. When 
the depositor wishes to draw upon his deposit, he makes 
out a check, of which the following is a common form : 



BANKING BUSINESS. 149 



(gfiicctyo, Jf,_ 



to tfiv oidei of 

f. 



A check is usually made payable to : 

1. " Self," in which case the drawer alone can collect it. 

2. The payee or bearer, or merely to "Bearer," in which cases 
any one can draw the money. 

3. The order of the payee, in which case the payee must indorse it. 

Most banks also receive money and issue Certificates of 
Deposit, of which the following is a common form : 



Certificate of Deposit 
FIRST NATIONAL BANK OF DETROIT. 

Detroit, Mich,, G^**<*z^ , -/9C>C>. 
r^a^&n. &++ k as deposited in this Bank 
<Lj**-u-e' &t4^Kz&.e^ f -- Doll 



ars 



. 

payable to **<*- order in current funds on the return of this Certificate 
properly endorsed, with interest at the rate of 3 per cent per annum if 
left three months, or 4 per cent if left six months. Interest hereon -will 
cease one year from date, 

>^>*. c^****'/^, Teller. <-^*'c^=z^-^c^Ue. v Cashier. 



Exercises. 1. Write a check for $54.75. 

2. Also one payable to the order of yourself, and properly indorse 
it so that it can be collected only by Richard Roe or his order. 

3. Write a certificate of deposit payable to your order, for $75, 
dated Jan. 4, drawing 3% if left 3 mo., or 3i% if left 6 mo. Compute 
its value on Nov. 19 ; also on May 1 1 ; also on Mar. 23. 



150 HIGHER ARITHMETIC. 

II. LENDING MONEY. 

If a person wishes to borrow money from a bank, and 
the bank is willing to lend it to him, he usually gives a 
promissory note. This note may be secured by depositing 
with the bank some evidences of value, as stocks, bonds, 
etc., usually known as " Collateral," or by having some 
responsible person indorse the note. At present banks 
frequently loan money without an indorser to persons of 
unquestionable financial standing, a custom formerly not 
common. Since the borrowing of money on an indorsed 
promissory note is the method most commonly followed, 
this is the only one here discussed. 

A bank note is usually in the following form : 



ftet date, 
to fray to tfo oidvi of ' J&o*^*s J%<^ f 73. 



at fy Jtnrt National Bank, Boston, 



Such notes are usually made payable in 1, 2, or 3 mo., or in 30, 
60, or 90 da., so that the bank can get its interest often, the interest 
then being reloaned. It was formerly the general custom to add 
three " days of grace," as mentioned on p. 141 ; but as already stated, 
a considerable number of states have abolished this custom. In the 
above note the words " without grace " make the note mature July 5; 
otherwise it would mature July 8, drawing interest to that date. 

As a rule no interest is specified in such notes, but interest is paid 
in advance and is called discount. The bank thus gets interest on 
interest, but this is allowed, in such cases, by law. 



BANKING BUSINESS. 151 

In the case of the above note, John Doe, the maker, wishes to 
borrow $75.00. He makes the note payable to the order of Richard 
Roe, with whose financial standing the bank is satisfied. Richard Roe 
indorses it, thereby promising to pay it if John Doe does not. The 
maker then takes it to the bank and receives $75.00 less the interest 
(or discount) on $75.00 for 2 mo. at the usual rate. Since Richard 
Roe indorses this note for the accommodation of the maker, he is 
called an accommodation indorser. 

On July 5, if John Doe does not pay this note, the bank places it 
in the hands of a Notary Public, who sends to Richard Roe, the 
indorser, a Notice of Protest. If this is not sent promptly, the 
indorser may assume that the note has been paid, and he is released 
by law. If this notice is placed in a properly addressed sealed 
envelope and deposited in the post office by the notary, the demands 
of the law have been fulfilled. The law of protest varies, however, in 
different states. 

In discounting notes, banks count the time by months or days 
according as the note specifies, and then compute the interest by the 
help of tables usually based on 360 da. to the year, calling 30 da. 1 mo. 

Exercises. 1. Are "days of grace" allowed by law in your 
state ? (If so, always add them hi solving the problems in this sec- 
tion ; otherwise not.) 

2. What is the day of maturity and the discount on the following 

notes : 

DATE. TIME NAMED. FACE. KATE OF DISCOUNT. 

(a) Apr. 1, 60 da., $250, 6%. 

(6) Oct. 17, 3 mo., $5000, 5%. 

(c) May 10, 90 da., $125, 7%. 

(d) Dec. 12, 2 mo., $50, 6%. 

(e) July 7, 4 mo., $600, 5%. 

3. What is the usual rate of discount on bank notes in your 
vicinity ? Using that rate, find the discounts on the following notes : 

DATE. TIME NAMED. FACE. 

(a) Apr. 15, 4 mo., $1000. 

(5) Jan. 3, 60 da., $500. 

(c) Aug. 5, 90 da., $750. 

(d) Dec. 9, 3 mo., $50. 

(e) Oct. 8, 2 mo., $75. 

4. Write and properly indorse bank notes subject to the conditions 
stated in Ex. 3. 



152 HIGHER ARITHMETIC. 



III. DISCOUNTING NOTES. 

Merchants frequently take notes from their customers, 
running 1, 2, or 3 mo. or even longer, and drawing inter- 
est. Such notes are often made payable at the bank in 
which the merchant keeps his account so that, in case he 
needs the money on a note before it is due, and sells it to 
the bank, the latter can the more easily collect it. In case 
of sale, the seller indorses the note and the bank discounts 
it ; that is, the bank pays the sum due at maturity, less 
the discount (interest) on that sum, this difference being 
called the proceeds. 

It will be seen that so far as the bank is concerned this process of 
discounting a note held by a customer is essentially that already 
described of lending money on an indorsed note. There are, how- 
ever, two differences : 

1. The indorser is not now an accommodation indorser ; he is the 
owner of the note and he receives the money from the bank. If, 
however, the maker does not pay the note when it becomes due it is 
protested like any other note and the indorser is held responsible as 
explained on p. 151. 

2. "The note usually draws interest and it frequently is not dis- 
counted on the day of its date. The discount is therefore reckoned 
on the face of the note plus the interest, or on the future worth, for 
the time between the day of discount and the day of maturity. This 
time is occasionally computed in exact days, but more often in months 
and days. 

E.g., a merchant takes from a customer a note for $755.50, dated 
Apr. 16, due in 90 da. without grace, at 6%. Needing the money 
on May 1 he indorses the note and deposits it in his bank. If the 
bank is discounting at 6%, it gives him credit for the proceeds. The 
computation is as follows : 

Face of note = $755.50 
Int. for 90 da. = 11.33 
Future worth = $766.83 
Disc't for 75 da. = 9.59 
Proceeds = $757.24 



BANKING BUSINESS. 



153 



Exercises. In the following problems take the rate of discount 
usually charged by banks in your vicinity, except as otherwise speci- 
fied, allowing days of grace or not according to their custom. The 
first exercise includes the practical business problems ; the rest are of 
value merely for the analysis. 

1. Find the discount and the proceeds on the following notes : 



FACE. 



DATE. 



TIME TO RUN. 



(a) $136.75, 


Feb. 7, 


3 mo., 


(6) $75.50, 


May 10, 


60 da., 


(c) $352.00, 


Oct. 5, 


2 mo., 


(d) $50.75, 


July 8, 


90 da., 


(e) $800.00, 


Jan. 10, 


4 mo., 


(/) $62.25, 


Dec. 8, 


3 mo., 



INTEREST. 


DISCOUNTED. 


7%, 


Apr. 1. 


6%, 


June 2. 


6%, 


Oct. 5. 


None, 


Sept. 1. 


5%, 


Feb. 10. 


None, 


Dec. 27. 



INT. DISCOUNTED. 


RATE OF 
DISCOUNT. 


6%, 


Mar. 4, 


7%. 


5%, 


July 20, 


6%. 


7%, 


Dec. 18, 


6%. 


None, 


Apr. 1, 


6%. 



2. Find the face of the following notes : 
PROCEEDS. DATE. TIME TO RUN. 

(a) $75.24, Feb. 4, 3 mo., g 
(6) $81.46, July 13, 2 mo., 6 

(c) $101.56, Oct. 3, 4 mo., | 

(d) $39.85, Apr. 1, 2 mo., f 

3. For how long is a note for $74.60 discounted at 6%, if the pro- 
ceeds are $73.85? 

4. At what rate is a note for $125.50 discounted for 4 mo., if the 
discount is $2.09 ? 

6. Do you know of any savings bank in your vicinity ? If so, 
what rate of interest does it pay and how often is this compounded ? 
Under these conditions, what would be the amount at the end of 
3 yrs, of $100 invested July 1 ? 

6. If you had a check on a bank in your vicinity, payable to your 
order, what would be the steps necessary to get the money ? What 
would be the steps necessary to transfer it to another person so that 
the money could be drawn only on his order ? 

7. Find the difference in the amount of $100 invested in a savings 
bank at 4%, compounded semi-annually, and the amount of the same 
sum at simple interest at 4%, the time being 5 yrs. in each case ; also 
for 6 yrs. ; also for 7 yrs. 

8. Which yields the better income on $100 in 10 yrs., 4%, com- 
pounded semi-annually, or 5% simple interest ? 

9. What principal will amount to $29,588.62 in 3 yrs. 3 mo. 3 da. 
at 6% ? 



CHAPTER XVII. 
Exchange. 



IF a person in one place owes a debt in another, he can 
settle it in a variety of ways. 

1. He may send the money 

(a) By an unregistered letter ; this is liable to be lost or stolen, 
although with our present postal service this liability is slight. 

(6) By a registered letter ; in case of loss this can be traced to the 
one at fault ; in case of loss by accident no recovery is possible, but 
otherwise the government, while not holding itself responsible, requires 
the one at fault to make good the loss. 

(c) By express or other messenger, in which case the company or 
messenger is liable for loss. 

2. He may cancel the debt by sending 

(a) A check on his home bank where he has money deposited, in 
which case the creditor may have to pay a bank for collecting it. 

(&) A draft drawn by some bank on a bank in a large city like New 
York or Chicago ; such a draft, especially if presented by a customer, 
is usually cashed without discount at any bank. 

(c) A postal money order ; this is not as safe as (a) or (&) since the 
government cannot be sued in case of payment to the wrong person ; 
identification is required, however, unless the sender waives it. 

(d) An express money order, issued by various express companies 
and costing the same as the postal order ; in case of loss or of payment 
to the wrong person these companies can be sued. 

(e) A telegraphic order ; this method is the most expensive, but 
the most rapid. 

The subject of Exchange relates to the second of these 
plans and includes the five methods named. 



EXCHANGE. 155 

(a) The check. This instrument has already been 
described on p. 149. 

If a check is drawn by John Doe of Albany on a bank in that city, 
payable to the order of Richard Roe of Cleveland, the latter on 
receiving it indorses it and deposits it in the bank where his account 
is kept. This bank will probably collect it for him without charge. 
This is the usual plan, and a large part of the indebtedness of the 
country is settled by checks. 

If Richard Roe has no bank account, the bank to which he takes 
the above check will require his identification, will charge him 
exchange, that is, a small sum for collecting it, and will probably not 
pay him the money until it has been received from Albany. 

(b) The draft. Drafts are usually in substantially this 
form : 



fast Iati0nal anh of Prang. 
fo ffo oidei of 



To Mercantile National Bank, \ 
New York aty. } 



It will be noticed that a draft is quite like a check, but 
it is signed by the cashier of some bank and is drawn on 
some bank in a large commercial center. 

In the case mentioned under (a), John Doe might have purchased 
a draft for the amount of his indebtedness, payable to his own order 
or to the order of Richard Roe ; if to his own order, he would have 
indorsed it payable to the order of Richard Roe. He might have 
to pay a slight premium to the bank, usually 10 cts. to 15 cts. for 
drafts under $100. On receipt of the draft, Richard Roe would 
indorse it and receive the money, usually without any discount, at a 
bank. 



156 HIGHER ARITHMETIC. 

The drafts already described are sometimes known as 
bankers' drafts to distinguish them from commercial drafts. 
The latter are extensively used by merchants, though 
rather as a means of demanding and collecting payment 
for a debt through the agency of banks than as a system 
of domestic exchange. 

The great majority of such drafts are substantially in 
the following form : 



fit bigfit feay to tfiv otdel of 






1. In the above case suppose John Doe has bought goods of 
Richard Roe to the amount of $53.75, say on 30 da. credit, and does 
not pay the bill when due. Roe may then make out a draft as above, 
payable to the order of his bank, and deposit it for collection. 

2. The Cleveland bank would send it to some Albany bank, asking 
it to collect and remit. 

3. The Albany bank would send a messenger to John Doe and 
demand payment. In some states 3 days of grace are allowed on a 
sight draft, though not in New York. In such case, or in case of a 
time draft (ie., a draft payable a certain number of days after sight), 
Doe writes "Accepted, July 8, 1898" (if that is the date) across the 
face and signs it ; at the proper time it is again presented by a mes- 
senger from the bank and payment is demanded. 

4. In case Doe declines to accept it, or to pay it if due immediately, 
the draft is returned to the Cleveland bank and Roe is notified ; he 
must then take other means for payment. 

5. If it is paid, the Albany bank remits by draft to the Cleveland 
bank, deducting a small amount for making the collection. 



EXCHANGE. 157 

The fluctuation of exchange. For small sums, say for 
$500 or less, New York or Chicago exchange always sells 
at a premium of about 0.1%. This is to pay the bank for 
its trouble and for the expense of shipping the money when 
its balance at New York or Chicago gets low. Banks 
usually buy New York or Chicago drafts at par, that is, 
at their face value, thus making no charge for cashing 
them. 

But on large sums the rate of exchange varies. If the 
San Francisco banks owe the New York banks $1,000,000, 
they must send that amount by express, an expensive pro- 
ceeding. If a man in San Francisco at that time wished 
to buy a draft on New York for $10,000 they would charge 
him more than usual because they would have to express 
that much more to New York. But if a man in New York 
wished to buy a draft on San Francisco he might buy it 
for $9999 or less because they would get their money at 
once and the risk and expense of transmitting it would be 
saved. 

The premium or discount is usually quoted as a certain per cent of 
the face of the draft, but sometimes as the amount on $1000. Thus, 
the quotation of % premium is the same as that of $2.50 premium. 

Exercises. 1. New York banks are selling drafts on New Orleans 
banks at 0.1% discount; which city is owing the other the more money? 
How much would a draft on New Orleans for $5000 cost in New York ? 

2. Denver is selling drafts on Boston at i% premium ; which city 
is owing the other the more money ? How much would a draft on 
Boston for $15,000 cost in Denver ? 

3. Suppose the balance of trade between Cincinnati and Chicago is 
said to be largely against Cincinnati, what does this mean ? In which 
place would the drafts on the other certainly be at a premium ? 

4. Suppose that in Denver drafts on New York are selling at \% 
premium, on Chicago at 0.1% premium, on San Francisco at 0.1% 
discount ; what is the probable balance of trade between Denver and 
each of the other cities ? 



158 HIGHER ARITHMETIC. 

The clearing house. If the draft shown on p. 155 is 
indorsed by John Doe payable to the order of Richard Roe 
of Cleveland, Roe will take it to his Cleveland bank to be 
cashed or placed to his credit. The Cleveland bank will 
send it to the New York bank with which it does business, 
say the Chemical National, and will receive credit for it. 
The next morning the Chemical National will send it to 
the Clearing House, where the leading New York banks 
send representatives to transfer the drafts held by each 
on the others. There it goes to the representative of the 
Mercantile National Bank on which it is drawn, and is 
paid. By the Mercantile National it is finally returned to 
the First National Bank of Albany which drew it. 

Since both the Chemical arid Mercantile National Banks 
have drafts on one another, and so for all other banks in 
the Clearing House, only a comparatively small balance is 
necessary to settle all accounts. 

Every large city has its own clearing house, but the one at New 
York is the largest in the country. In fact, its exchanges aggregate 
more than those of all the others together, averaging about $100,000,000 
daily. The banks do not pay the balances to one another ; but if a 
bank owes a balance of $100,000 to all the others, it pays this to the 
clearing house; and if another bank has a balance in its favor of 
$50,000, the clearing house pays it that sum. In this way, the amount 
coming into the clearing house must equal the amount to be paid out. 

Exercises. 1. The exchanges in the New York Clearing House 
for one week were $623,405,190, and the actual balances paid were 
$36,951,619 j what was the average of each per day, and the balances 
for the week were what per cent of the exchanges ? 

2. Since 1880 the highest average daily clearings for any one year 
at the New York Clearing House were $159,232,191 for 1881 ; the 
average daily balance paid in money was 3.5%; how much was 
this? 

3. In the same period the lowest average daily balance paid in 
money was $4,247,069, which was 5.1% of the average daily clearings ; 
to how much did the average daily clearings amount ? 



EXCHANGE. 159 

(c) The postal money order is substantially the same as 
a draft, except that instead of being drawn by a bank 
cashier on a bank in some large city it is drawn by one 
postmaster on another. These orders are always sold at a 
premium and cashed at par. The premium (price) varies 
from 3 cts. to 30 cts. depending on the amount, which may 
be from 1 ct to $100. 

(d) The express money order is substantially like the 
postal money order. 

(e) The telegraphic money order. Telegraph companies 
receive money at one office and telegraph (usually through 
the office of the manager of this part of the business) to 
another office to pay out an equal sum to the person 
named. A higher fee is charged than for drafts, but this 
method is employed when great promptness is necessary. 

Exercises. 1. If you owed $25 in Syracuse, N. Y., which of the 
five methods named would you take to pay it ? Why ? 

2. What would be the cost of a $500 draft at 0.1% premium ? at 
par ? at 0.1% discount ? 

3. If the fee for a money order over $75 and not exceeding $100 
is 30 cts., what is the total cost of a money order for $87.50 ? 

4. To send money by telegraphic order costs the double rate for a 
10- word message and 1% of the sum sent ; if the rate for a 10-word 
message is 40 cts. , what would be the total cost of a telegraphic order 
for $87.50 ? 

5. When a New York draft for $40,000 can be bought in St. Louis 
for $39,950, is exchange at a premium or a discount ? What is the 
rate ? How is the balance of trade ? 

6. Find the cost of each of the following drafts : 

FACE. EXCHANGE. FACE. EXCHANGE. 

(a) $4350, i%prem. (/) $1276.90, i% prem. 

(6) $9275, % disct. (g) $2493.60, par. 

(c) $2450, 1% (h) $4275.75, i% prem. 

(d) $7500, $1.25 prem. (i) $10,023.60, $1 disct. 

(e) $8556.75, 75 cts. disct. 0') $9870, $1.50 prem. 



160 HIGHEB, ARITHMETIC. 

7. The cost of a draft including the premium of 0.1% is $4254.25 ; 
what is the face ? (Business men usually merely subtract the premium 
from the cost, a process which, while not accurate, gives a sufficiently 
close result on sums below $10,000. In this and the following exer- 
cises find both the correct result and the business approximation.) 

8. Find both accurately and by the business approximation the 
face of each of the following drafts (see Ex. 7) : 

COST. EXCHANGE. COST. EXCHANGE. 

(a) $5244.75, $1 disct. (d) $5012.50, i% prem. 

(6) $1757.80, i% " (e) $4268.07, 25 cts. " 

(c) $13,593.29, i%prem. (/) $14,518.13, $1.25 " 

Foreign exchange is subject to the same general laws as 
domestic exchange, differing chiefly as to the currency and 
the manner of quoting the rate of exchange. 

Thus, the par of exchange on London is 4.8665, that is, 1 in gold 
is worth $4.8665 in gold. If exchange is selling at 4.90 it is above 
par, a draft for 1 costing $4.90 ; while if it is selling at 4.84 it is 
below par. Exchange on London and other cities in Great Britain 
and Ireland is always quoted at so many dollars to the pound. 

Exchange on Paris, and other cities in France and in countries 
like Belgium and Switzerland which use the French monetary sys- 
tem, is usually quoted at so many francs to the dollar, the quotation 
5.14 meaning that $1 will buy a draft for 5.14 francs. It is some- 
times, and more conveniently, quoted at so many cents to the franc, 
the quotation 19.8 meaning that 19.8 cts. will buy a draft for 1 franc. 
The par of exchange is about 5.18, or 19.3. 

Exchange on German cities is usually quoted at so many cents to 
4 marks, the quotation 96 meaning that 96 cts. will buy a draft for 
4 marks. It is sometimes, and more conveniently, quoted at so many 
cents to the mark, the quotation 23 meaning that 23-J cts. will buy a 
draft for 1 mark. The par of exchange is about 95.2, or 23.8. 

Foreign drafts are usually called bills of exchange, a bill 
at 30 days' sight being a draft due 30 days (or 30 days 
+ 3 days of grace) after sight. 

It is the custom with foreign bills, and occasionally with 
domestic drafts (sometimes called inland bills) between 
distant cities, to make out duplicates, as follows : 



EXCHANGE. 161 



50. New York [date] , No. 147638. 

At sight of this first of exchange (second of the 
same tenor and date unpaid) pay to the order of 
Brown Brothers & Co. fifty pounds sterling, value 
received, and charge the same to the account of 

To Brown, Shipley & Co., "1 

' f J j r John Doe. 

London, Jbmgland. J 



50. New York [date] , No. 147638. 

At sight of this second of exchange (first of the 
same tenor and date unpaid) pay to the order of 
Brown Brothers & Co. fifty pounds sterling, value 
received, and charge the same to the account of 

To Brown, Shipley & Co., "1 

T j -ci i j f Jonn 

London, England. J 



Such duplicate drafts form a set of exchange. Formerly three 
such drafts were made out ; at present the leading dealers in foreign 
exchange draw their bills in duplicate ; recently, in the case of 
express company drafts used by tourists, only a single bill is 
demanded, and the custom is extending. 

Exercises. 1. What is the cost of a draft on London for 100, 
exchange 4.90 ? Is the balance of trade, judged by this quotation, 
against this country or against England ? 

2. What is the cost of a draft on Paris for 1000 francs, exchange 
5.20 ? exchange 5.13 ? Which result is the greater ? In which case 
is the balance of trade against this country ? In which is it in favor 
of this country ? 

3. What is the cost of a draft on Paris for 1000 francs, exchange 
19.8 ? exchange 19 ? Which result is the greater ? In which case 
is the balance of trade against this country ? In which case is it in 
favor of this country ? 

4. What is the cost of a draft on Leipzig for 500 marks, exchange 
97 ? exchange 94 ? Consider the balance of trade in each case. 



162 HIGHER ARITHMETIC. 

5. What is the cost of a draft on Hamburg for 200 marks, exchange 
23| ? exchange 94 ? 

6. Find the cost of each of the following drafts : 

FACE. DRAWN ON. BATE OF EXCHANGE. 

(a) 700, London, 4.84. 

(&) 2750 francs, Paris, 5.19. 

(c) 6280 marks, Frankfort, 95. 

(d) 525 8 shillings, Liverpool, 4.88. 

(e) 1425 francs, Paris, 19. 
(/) 800 marks, Berlin, 23f. 
(g) 25 4 shillings, London, 4.90. 
(h) 750 francs, Brussels, 5.18. 
(i) 575 marks, Leipzig, 24. 
(j) 50, Glasgow, 4.87. 
(k) 8760 marks, Munich, 95f 

7. A New York merchant owes the following sums to foreign 
dealers ; if he remits by draft, what is the face in each case ? 

AMOUNT OWED. DRAFT PAYABLE AT. BATE OF EXCHANGE. 

(a) $2435, London, 4.87. 

(6) $1920, Paris, 5.20. 

(c) $2400, Leipzig, 96. 

(d) $1958, Liverpool, 4.89|. 

(e) $81.62, Paris, 19. 
(/) $117.50, Berlin, 23f 

8. An express company sells travelers' checks payable in various 
countries of Europe. It charges \% premium, and every $20 check 
allows the owner to draw 4 Is. 6d. in England, or 102.50 francs in 
France, or 82.50 marks in Germany. The company also has the use 
of the money until the checks are paid, an average of 2 mo., the use 
of the money being worth at the rate of 5% a year. On $500 of checks, 
paid in England, how much does the company make above the par of 
exchange ? 

9. In Ex. 8, suppose the checks paid in Germany. 

10. In Ex. 8, suppose the checks paid in France. 

11. The rates for foreign money orders, payable in the currency 
of the country to which they are sent, are : for sums not exceeding 

$10, 10 cts.; $10-$20, 20 cts.; $20-$30, 30 cts.; $90-$100, $1. 

What would be the cost of the following money orders : 

(a) $75 payable in London ? 
(6) $62.50 " Paris? 



CHAPTEE XVIII. 
Government Revenues. 



CERTAIN revenues are necessary for the support of the 
governments of the United States, the various individual 
states, the counties, the cities, etc. The methods of 
obtaining these revenues are prescribed by law and vary 
for these different kinds of governments. 



I. THE UNITED STATES GOVERNMENT. 

The expenses of our general government are about a 
million dollars a day, and our income should be about the 
same or enough more to gradually reduce our indebtedness. 
Some of our sources of income and our principal expendi- 
tures are as follows, although all of the items vary from 
year to year : 



INCOME. 
Internal revenue 

Spirits $90,000,000 

Tobacco $30,000,000 

Fermented liquors $32,000,000 
Oleomargarine and 

other penalties $2,000,000 
Customs revenue, 

$160,000,000 to $200,000,000 

Total, including the above and the 

income from public lands, etc., 

$325,000,000 to $400,000,000 



EXPENDITURES. 

War dept. $50,000,000 

$30,000,000 
$10,000,000 
$135,000,000 
$30,000,000 



Navy " 
Indians 
Pensions 
Interest on debt 
Diplomatic and consular service, 
and miscellaneous, $100,000,000 
Total,$325,000,000 to $400,000,000 



164 HIGHER ARITHMETIC. 

The customs revenue (tariff, duty) is collected at custom 
houses situated at ports of entry established by law. 

Merchandise brought into the country (1) is on the free 
list (i.e. } it is not subject to duty), or (2) is subject to ad 
valorem duty .(a certain per cent on the value at the place 
of purchase), or (3) is subject to specific duty (a certain 
amount by number, measure, etc.), or (4) is subject to both 
ad valorem and specific duty. 

E.g., by the tariff of 1883 apples were on the free list; by the 
tariff of 1890 they paid a specific duty of 25 cts. a bushel ; by the 
tariff of 1894 they paid an ad valorem duty of 20%. By the tariff of 
1890 oriental rugs paid a specific duty of 60 cts. per sq. yd., and an 
ad valorem duty of 40%. 

Ad valorem duty is, if honestly collected, the more fair ; but on 
account of undervaluation by the importer there is much more chance 
for fraud in the collection. 

Exercises. 1. Taking the total revenue of our general govern- 
ment for one year as $326,926,200, and our internal revenue as 
$146,762,865, what per cent of our income was of this class ? 

2. In one year the internal revenue was $143,421,672, including 
$79,862,627 from spirits, $29,707,908 from tobacco, and $31,640,618 
from fermented liquors ; what per cent was derived from these three 
classes ? 

3. The revenue of the post office department for a certain year was 
$76,983,128, and the expenditures were $86,790,172 ; the excess was 
what per cent of the revenue ? 

4. Mathematical instruments pay a duty of 35%; what is the 
invoice price of an instrument which pays a duty of $6.30 ? 

5. The duty on cheese is 4 cts. per Ib. ; how much does a city which 
consumes 40,000 Ibs. of French cheese a year pay to the government 
for this privilege ? 

6. The duty on aniline dyes being 25%, what is the valuation at 
the custom house on a package of dyes which pays $59.38 ? 

7. The duty on fine blankets being 35%, what is the invoice price 
of a shipment of blankets which cost the importer $786.73, including 
the duty and $12.50 freight ? 

8. Rubber coats pay a duty of 40% ; how much is the duty on 
100 doz. invoiced at 1 Is. a dozen, reckoning the pound at $4.86f-? 



GOVERNMENT REVENUES. 165 

9. Ready-made woolen clothing pays a duty of 50% ; how much 
less would a $20 suit cost if it were on the free list, not considering 
the freight and profit ? 

10. Cutlery valued from $1.50 to $3 a dozen pays a duty of 75 cts. 
a dozen and 25% ad valorem ; what is the duty on 100 doz. Sheffield 
knives invoiced at $2.25 a dozen ? 

11. English books pay a duty of 25%; how much less would you 
have to pay for an English book which costs you $8, including the 
duty and 50 cts. postage, if it were not for this tariff ? 



II. STATE AND LOCAL TAXES. 

The method of collecting taxes varies in different states, 
but in general it may be said that a valuation is placed 
upon the property of corporations, of land owners, and of 
persons possessing any considerable amount of personal 
property. Upon this assessed valuation a certain rate of 
taxation is fixed. 

The expression rate of taxation is usually applied to the 
number of mills of tax on each dollar of valuation. 

Thus, if the rate is 5-J mills, the tax is 5^- mills on each dollar. 

The rate of taxation is, therefore, found by dividing the 
amount to be raised by the number of dollars of valuation. 

E.g., if a village has to raise $12,575, and if the valuation is 
$2,465,685, the rate of taxation is ^f^ = $0.0051 on $1. 

In addition to the tax already mentioned, male citizens 
over 21 years of age are frequently required to pay a poll 
(i.e., head) tax. 

If taxes are not paid when due a fine is usually imposed 
in the form of a certain per cent of increase of the tax. 

E.g., if a man's taxes are $12, and he does not pay them when due, 
the law may require him to pay 5% additional, thus making his tax 
$12.60. 



166 



HIGHER ARITHMETIC. 



Tax collectors usually prepare a table similar to that 
given below. For this table, the rate of 5-J- mills on $1 
has been taken. 



TAX TABLE. KATE 5 MILLS ON $1. 






1 


2 


3 


4 


5 


6 


7 


8 


9 


0000 


0055 


0110 


0165 


0220 


0275 


0330 


0385 


0440 


0495 


0550 


0605 


0660 


0715 


0770 


0825 


0880 


0935 


0990 


1045 


1100 


1155 


1210 


1265 


1320 


1375 


1430 


1485 


1540 


1595 


1(550 


1705 


1760 


1815 


1870 


1925 


1980 


2035 


2090 


2145 


2200 


2255 


2310 


2365 


2420 


2475 


2530 


2585 


2640 


2695 


2750 


2805 


2860 


2915 


2970 


3025 


3080 


3135 


3190 


3245 


3300 


3355 


3410 


3465 


3520 


3575 


3630 


3685 


3740 


3795 


3850 


3905 


3960 


4015 


4070 


4125 


4180 


4235 


4290 


4345 


4400 


4455 


4510 


4565 


4620 


4675 


4730 


4785 


4840 


4895 


4950 


5005 


5060 


5115 


5170 


5225 


5280 


5335 


5390 


5445 



The column at the left gives the first figure of the number of 
dollars of valuation, and the row at the top the second figure. A 
decimal point is understood before each of the other numbers. 
E.g., the tax on $10 is $0.055, 

" $87 " $0.4785, 

" $5900 " $32.45. 

To find the tax on $9805, the collector's commission being 1%, the 
actual computation of a collector would be as follows : 
Tax on $9800 = $53.90 

" $5 - .03 

$53.93 

Commission 1% = .54 
$54.47 

Exercises. 1. From the preceding table find the tax at 5-J- mills 
on $1 on each of the following valuations, the collector's commission 
being 1% : (a) $1750, (b) $2500, (c) $5475, (d) $17,645, (e) $18,750, 
(/) $9250, (g) $7625. 

2. Prepare the first two rows of a tax table (opposite and 1) at 
the rate of 8| mills on $1. 



GOVERNMENT REVENUES. 167 

3. From Ex. 2, compute the tax on each of the following valua- 
tions at 8-J- mills on $1, collector's commission being 1%: (a) $1200, 
(6) $19,150, (c) $15,175, (d) $1750, (e) $17,150, (/) $1825, (g) $825, 
(h) $500. 

4. Taxes are levied in a certain village as follows : for streets" 
$2000, for fire apparatus, etc. $1500, for school purposes $6000, for 
salaries and office rent $3400, for repair of bridge $500, for general 
purposes $500 ; the total valuation of property is $1,950,000 ; what is 
the rate of taxation ? 

5. In Ex. 4, what would be the taxes of a man whose property is 
valued by the assessors at $2500 ? 

6. The rate of taxation being 5 mills on $1, what are the taxes on 
property valued by the assessors at $9500, collector's commission 1%? 
Suppose the owner does not pay promptly and is fined 5%, what is his 
tax? (Use the table.) 

7. Find the tax on 

(a) $18,500 at 4 mills on a dollar, 

(b) $6000 "3.8 " 

(c) $3500 "8.4 " 

(d) $21,400 " 7.2 " 

(e) $5500 "5 

(/) $6750 " 4.8 " 

(g) $1800 " 54- 

8. The assessed valuation of a district being $950,725, what is the 
rate of taxation necessary to raise $8000 ? 

9. To raise $2000, a tax of 1 mills on a dollar was levied ; what 
was the assessed valuation ? 

10. What would be the various taxes levied on a man whose 
property is valued by the assessors at $12,800, if the rates were as 
follows : state tax 1-J mills, county 2 mills, town 0.8 mill, school 
1.4 mills ? 

11. At 7 mills on a dollar, how much is the tax of a man who 
owns a farm of 250 acres, worth $70 an acre, but assessed for only f 
of its value ? 

12. The rate of taxation in a certain town is 5 mills on a dollar, 
and the amount to be raised is $4783.87 ; what is the assessed valua- 
tion ? 

13. At 6 mills on a dollar, how much is the tax of a man who 
owns a farm of 300 acres assessed at $10 an acre, and who is assessed 
on $2000 of personal property, and who pays a poll tax of $1 ? 



CHAPTER XIX. 
Commission and Brokerage. 



PRODUCE bought in quantities or sent to cities for sale 
is usually bought or sold through a commission merchant or 
a broker. 

A commission merchant usually has the goods consigned 
to him and sells them in his own name, remitting the net 
proceeds (the sum realized less the commission) to the con- 
signor. If he is buying for a customer, he charges the sum 
paid plus his commission. 

A broker does not receive the goods, but sells them for 
the consignor in advance or buys them for his customer, 
and they are shipped directly to the buyer. His commis- 
sions, called brokerage, are therefore less than those of the 
commission merchant. 

Commission and brokerage are reckoned as a certain per 
cent of the amount paid in buying or realized in selling, but 
more often as a certain amount for a given transaction. 

Thus, it is more common to pay 50 cts. a ton for selling hay than 
to pay a commission of 4% or 5%. Stocks are bought and sold on a 
brokerage of 12 cts. for each share, as explained in Chap. XX. 

There are numerous other cases involving commission 
and brokerage, as the buying and selling of securities. 
Some of these are mentioned in the exercises. 

Since no new principles are involved, illustrative problems are 
unnecessary. 



COMMISSION AND BROKERAGE. 169 

Exercises. 1. What is the commission for buying a carload of 
400 bu. of grain at of a cent a bushel ? for selling 4 carloads of hay 
at $5 a car ? for selling 500 bu. of beans at 95 cts. a bushel, commis- 
sion 5% ? 

2. How much does a broker receive for selling 1200 bales of cotton, 
brokerage 25 cts. a bale ? 500 bbls. of rye flour at $2.95, brokerage 
2i% ? 10,000 bu. wheat, brokerage (of a cent a bushel) ? 

3. A commercial traveler sells goods at a commission of 3% ; to 
how much must his sales amount that he may have an income of 
$4500 a year ? 

4. A commission merchant receives 100 boxes of Mexican oranges 
which he sells at $3.50 a box and remits $323.80 net proceeds; what 
is the rate of his commission ? 

5. What are the net proceeds of a sale of 8750 Ibs. of leather at 
25 cts. a pound, commission 2% ? 

6. A speculator buys 1000 bbls. of May pork (ie., to be delivered 
the following May) at $7.82^-, and sells it at $7.90; he pays a bro- 
kerage of 2 cts. (on each barrel) for buying and the same for selling ; 
does he gain or lose, and how much ? 

7. A speculator buys 10,000 bu. of May wheat at 83 (cts. a 
bushel) and sells it at 82 ; the brokerage is (of a cent a bushel) for 
buying and the same for selling ; how much does he lose ? 

8. An auctioneer offers his services at $8 a day or 2% of amount 
sold ; a merchant accepts the latter offer and the stock is disposed of 
in 4 da., realizing $1875.50 ; how much less would he have paid if he 
had taken the first offer ? 

9. A collector has a $500 note placed in his hands with power to 
compromise ; he accepts 75 cts. on a dollar and charges 5% of the sum 
collected, and 25 cts. for a draft ; what are the net proceeds ? 

10. A broker buys flour for a customer at $3.30 a barrel, charging 
2% ; the bill, including commissions, is $4039.20 ; how many barrels 
are bought ? 

11. A dealer buys 1000 doz. eggs at an average price of 16 cts. a 
dozen and sends them to a commission merchant who sells them at 
an advance of 4 cts. a dozen, charging 10% commission ; the express 
was $7.50 ; did the dealer gain or lose, and how much ? 

12. At 5%, what is the brokerage for selling 1000 bu. of potatoes 
at 38 cts. a bushel ? 

13. A commission merchant sells 275 bu. of onions at 60 cts. a 
bushel, and remits the proceeds after deducting his commission of 
7i what is the amount remitted ? 



170 HIGHER ARITHMETIC. 

14. A commission merchant remits $266 as the proceeds of a sale 
of 200 bbls. of apples, his commission being 5% ; at what price per 
barrel did he sell them ? 

15. A man sends a carload of 13 tons of hay to Boston where it 
sells for $14 a ton, and receives $175.50 after paying his broker ; how 
much was the brokerage a ton ? 

16. In Ex. 15, if the hay cost the man $8.50 a ton, and the freight 
cost 21 cts. a 100 (Ibs.), did he make or lose by the transaction, and 
how much ? 

17. A commission merchant sells 4000 heads of cabbage at $3.50 a 
hundred, and remits $126 ; what was his rate of commission ? 

18. 400 bu. of beans at 62 Ibs. to the bushel are shipped to Boston, 
the freight being 28 cts. a 100 (Ibs.) ; the beans cost the shipper 70 
cts. a bushel and were sold through a broker at 95 cts., brokerage 5% ; 
how much did the shipper gain ? 

19. A commission merchant sold 600 Ibs. of butter at 24 cts., 
480 doz. eggs at 20 cts., 1200 Ibs. poultry at 7 cts.; what are the net 
proceeds after deducting $14 for freight and cartage and 2-$-% com- 
mission ? 

20. A broker remits $1706 after deducting 2% for brokerage and 
25 cts. for the draft ; how much was his brokerage ? 

21. A salesman received $6782.88 in one year, this representing 
his commissions at 1J% ; find the amount of his sales. 

22. A lawyer having a debt of $3250 to collect, compromises for 
97^- cts. on a dollar ; his commissions are 2 J% ; how much does he 
remit to his client ? 

23. A lawyer collects a debt for a client, takes 3J% for his pay, and 
remits the balance, $1935 ; what was the debt and the fee ? 

24. An agent buys goods on commission at 2|%, and pays $40 for 
freight ; the whole amount was $1628.73 j what was the sum expended 
for goods ? 

25. A real estate agent sold some western land for a man and, after 
retaining $23.40 as his commission, remitted $2116.60 ; what rate of 
commission did he charge ? 

26. An agent sells some property for s dollars on a commission of 
r% ; what are the net proceeds ? 

27. The net proceeds from the sale of some property is p dollars, 
and the rate of commission is r% ; at what price was it sold ? 

28. An agent sells some property for s dollars and remits p dollars 
as the net proceeds ; what was the rate of commission ? 



CHAPTER XX. 
Stocks and Bonds. 



WHEN a number of persons wish to engage in business 
the law allows them to form a corporation usually known 
as a stock company with a certain capital stock, each person 
owning a certain number of shares of that stock, each share 
being allowed one vote at the meetings for the election of 
directors. The business of these companies is managed by 
officers, usually elected by the directors. 

If the company makes more than its expenses, part or 
all of the surplus is divided among the stockholders in the 
form of dividends. 

If a stock is paying a good rate of dividend, that is, a 
higher rate than can be received from ordinary invest- 
ments, a $100 share will cost more than $100, and the 
stock is said to be above par. If it is paying about the 
same rate that ordinary investments bring, a $100 share 
may be bought for $100, and the stock is said to be at 
par. If it is paying low dividends, or none, it will be 
below par. 

The dividends are expressed either as a certain per cent of the par 
value or as a certain number of dollars per share. E.g., a 5% stock is 
one which is paying 5% on the par value ; and if stock is paying a 
dividend of $3 it pays $3 on each share. If the par value of one share 
is $100, then a stock paying $3 a share is the same as a 3% stock. But 
if, as is often the case with mining stocks, the par value is $25, a 
dividend of $3 a share is at the rate of 12%. 



172 HIGHER ARITHMETIC. 

Sometimes a company issues two kinds of stock, pre- 
ferred, which is entitled to the dividends to a certain 
amount (e.g., to 5% of the par value), and the common, 
which is entitled to part or all of the balance. 

On Jan. 1, 1897, the total capital stock of all steam railways in 
North America was $5,008,352,237, of which $3,986,753,937 was 
common stock and $1,021,598,300 was preferred. 

When a company needs more money than has been paid 
in by the stockholders it often borrows money and issues 
bonds payable at a certain time and bearing a certain rate 
of interest. 

These bonds are usually secured by a mortgage on the property of 
the company, taken in the name of trustees for the bondholders. 

Similarly, when a national, state, county, or city government 
wishes to borrow money it issues bonds, but without mortgages. 

Bonds either have coupons annexed, which are cut off as interest 
becomes due and are collected for the owner by the bank where he 
keeps his account, or are registered, that is, bear the name and address 
of the owner, the interest being sent when due. 

Bonds are spoken of as "4's reg.," " 5's coup.," etc., meaning that 
they draw 4% of their par value and are registered, or draw 5% of 
their par value and have coupons annexed. 

Exercises. 1. "Which would you prefer to own, common stock or 
preferred stock ? Why ? Suppose the preferred stock paid 5% and 
the common 7% ? the common 3% ? 

2. Suppose the capital stock of a company is $100,000, half being 
preferred and half common, the former being entitled to 5% and the 
latter to the balance ; suppose $6000 to be distributed in dividends, 
what rate of dividend would be received by the common stock ? If 
the dividends remain the same from year to year, which kind of stock 
would you prefer to have at the same price ? 

3. A company having $100,000 capital, of which $30,000 is pre- 
ferred stock entitled to 5%, the balance going to the common stock, 
has $6000 available for dividends ; what is the rate of dividend of the 
common stock ? 

4. Which would you prefer to own, $1000 of stock in a certain 
railway, or one of its $1000 bonds ? Suppose it was a 5% bond, while 
the stock paid 7% ? 5% ? 4% ? 



STOCKS AND BONDS. 173 

5. Which would you prefer to own, a coupon bond or a registered 
bond ? Why ? Which is the safer against loss by theft ? Which is 
the more easily transferred in case you wish to sell ? 

6. A certain railway stock is paying 9% dividends annually, and 
another is paying 2% ; are they above or below par ? Why ? 

7. United States 4% bonds are sold at 117, that is, a $100 bond 
costs $117, while Atchison railway 4% bonds are sold at 80 ; what is 
the reason for this difference in price ? 

8. In 1896 $68,981,244 of dividends was paid on $3,986,753,937 
of common stock in the North American railway companies, and 
$16,533,019 on $1,021,598,300 of preferred stock; what was the 
average dividend in each case ? 

Purchasing stocks and bonds. Since one usually does 
not know who has stock for sale he applies (directly or 
through a bank) to a stock broker who belongs to some 
stock exchange, where stocks and bonds are bought and 
sold. The leading stock exchange of the United States 
is in New York. 

The broker charges brokerage, usually -J% of the par 
value. This is charged for buying and also for selling. 

A newspaper quotation of 122 means that $100 of stock, which we 
shall always take as representing the par value of one share, as is 
usually the case with railway stocks, is selling for $122. But the 
seller would receive only $122 $&, or $121, for each share, because 
he must pay his broker ; and the buyer must pay $122'+ $, or $122$-, 
for each share, because he too must pay his broker. 

In stock quotations, fractions are expressed in eighths, quarters, or 
halves. E-g>, stock is often quoted at 97-f, but never at 62f. 

Fractions of a share are not usually sold ; if a person has $1000 to 
invest in Canada Southern Railway stock, quoted at 48|-, he would 
pay $49 a share, and purchase 20 shares and have $20 left. 

The purchaser receives a certificate of stock, signed by 
the proper officers of the company, stating that he owns 
so many shares. When he sells his stock he sends this 
certificate, properly indorsed, to his broker ; it is delivered 
to the company and another certificate is made out for the 
new purchaser. 



174 HIGHER ARITHMETIC. 

Newspaper quotations of the prices of stocks and bonds 
are given in the daily papers and form the best basis for 
a series of problems. The brokerage must be considered 
in each case. In the absence of a daily paper the following 
quotations may be used, and on them are based the prob- 
lems on pp. 175, 176. 



STOCKS. 




BONDS. 




Atchison 


16* 


U. S. 4's reg. 


116| 


" prefd. 


26f 


U. S. 4's coup. 


117 


C. B. & Q. 


79f 


U. S. 2's 


95 


C. & N. W. 


104J 


Atchison 4's 


80 


Canada South. 


51 


Bait. & Ohio 5's 


99 


N. J. Central 


107i 


Erie 7's 


137 


Canadian Pacif. 


57 


North. Pac. 6's 


114* 


D. L. & W. 


160} 


Wabash 5's 


107* 


Lake Shore 


148* 


N. J. Central 5's 


118i 


N. Y. Central 


97 


111. Central 4*'s 


llOi 


Pullman Car Co. 


158 


C. B. & Q. 5's 


100 



Illustrative problems. 

1. Suppose a man buys 10 shares of Atchison as quoted above and 
sells them 6 mo. later when quoted at 18, having received no divi- 
dends ; does he gain or lose, and how much, money being worth at 
the rate of 4% a year to him ? 

1. He buys for 16* + fc, and sells for 18 fc, 
.-. he gains 1, that is, $1.25 on a share. 

2. .-. " 10 -$1.25, or $12.50. 

3. But he loses * of 4% of 10 ($16* + $i), or $3.33, interest. 

4. .-. his net gain is $12.50 $3.33, or $9.17. 

2. Suppose a man buys 50 shares of C. & N. W. as quoted above 
and sells them 6 mo. later when quoted at 102*, meanwhile receiving 
a 3% dividend ; does he gain or lose, and how much, money being 
worth at the rate of 5% a year to him ? 

1. He loses ($104* + $i) ($102* $*), or $2, on a share. 

2. .-. " 50 -$2= $100. 

3. He also loses 2*% of 50 ($104* + $i) = $130.47, interest. 

4. .-. his total loss is $100 + $130.47 = $230.47. 

5. He gains 3% of 50 $100 = $150. 

6. .-. his net loss is $230.47 $150 = $80.47. 



STOCKS AND BONDS. 175 

3. Not considering the length of time the bond runs, what rate of 
income does a purchaser receive from investing in U. S. 4's reg. as 
quoted on p. 174 ? 

1. He receives $4 on every ($116 + $i) invested. 

2. Let r% stand for the rate. 

3. .-. r% of $116|- = $4. 



4 - '** =&=**>* 

Exercises. Unless otherwise directed, use the quotations given 
on p. 174, remembering the brokerage in each case. In finding the 
rate of income on bonds the time of maturity is not considered in 
these exercises. 

1. What will 20 shares of Pullman Car Co. stock cost ? 

2. Also 125 shares of N. Y. Central ? 

3. Also 75 shares of Lake Shore ? 

4. What sum will be received from the sale of 10 shares of C. 
B. & Q. ? 

5. Also from the sale of 40 shares of D. L. & W. ? 

6. Suppose a man buys 100 shares of N. J. Central as quoted and 
sells it when quoted at 115f, what is the gain, not considering divi- 
dends or interest ? 

7. Solve Ex. 6, supposing the stock had paid a 2% dividend mean- 
while, and that 8 mo. had elapsed and that money was worth at the 
rate of 6% a year to the investor. 

8. Suppose a man sells 100 shares of Canada Southern as quoted, 
this stock paying 2|% dividends annually, and invests the proceeds in 
31 shares of I). L. & W. which pays 9% dividends annually, putting 
the balance in a savings bank where it draws 4% ; find the alteration 
in income. 

9. Suppose a man sells 50 shares of C. B. & Q., which pays 4% 
dividends, and invests the proceeds in Lake Shore, which pays 8%, 
buying as many shares as possible, and placing the balance in a 
savings bank where it draws 4% ; find the alteration in income. 

10. Suppose a man has $2500 to invest ; what is the greatest num- 
ber of shares of Pullman Car Co. that he can buy, and how much 
will he have left ? 

11. Which investment pays the better, a 5% bond and mortgage or 
Erie 7's as quoted, the interest being paid promptly ? 

12. Also Erie 7's or North. Pac. 6's ? 

13. Also U. S. 2's or U. S. 4's coup. ? 



176 HIGHER ARITHMETIC. 

14. Also C. B. & Q. 5's or North. Pac. 6's ? 

15. A man's income in Erie 7's is $245 ; how much has he invested, 
at par value ? How much did the bonds cost him, as quoted ? 

16. A man's income in D. L. & W. stock, while it pays 9% annu- 
ally, is 6% on the sum invested ; what was the quotation when he 
made the investment ? 

17. A man's income is 5f% on the sum invested in C. & N. W. 
which he purchased when quoted at 104 ; find the rate of dividends. 

18. A broker bought on his own account 50 shares of Atchison 
prefd. as quoted and sold it at 2?i ; how much did he gain ? 

19. Tamarack Mining Co. stock pays a semi-annual dividend of 
3% ; how much will the holder of 50 $100-shares receive ? 

20. A bank with a capital of $150,000 declares a quarterly dividend 
of 2% ; what is the total amount of this dividend and how much will 
the owner of $1200 of stock receive ? 

21. To raise more money a company sometimes assesses its stock- 
holders. If a certain mining company levies an assessment of 10%, 
how much must be paid by the holder of 50 $100-shares ? 

22. How much must be invested in Wabash 5's as quoted to bring 
an annual income of $1000 ? 

23. The common stock of a certain railway company is $20,000,000, 
and the preferred stock (which in this case is entitled to 6% annually) 
is $4,000,000. The company declares a semi-annual dividend, paying 
the usual amount to the preferred stockholders and 2 \% to the others. 
How much money was distributed in dividends ? 

24. How much must be invested in 111. Central 4's as quoted to 
bring an annual income of $1350 ? 

25. How much income will be derived from an investment of 
$991.25 in Bait. & Ohio 5's as quoted ? 

26. A certain stock is quoted at 260 ; a broker is instructed to buy 
a certain number of shares at this price ; his bill including brokerage 
is $2081; how many shares did he buy ? 

27. The average rate of dividends paid to stockholders in national 
banks in 1872 was 10.19%, the dividends amounting to $46,687,115; 
in 1895, 6.96%, amounting to $45,969,663 ; what was the total capital 
for each of these years ? 

28. In Ex. 27, find the rate of increase of capitals and the rate of 
decrease of dividends. 

29. Of the bonds quoted on p. 174, which yields the highest rate of 
income on the investment ? 

30. Of the same bonds, which yields the lowest rate of income ? 



CHAPTER XXI. 
Insurance. 



FOB the majority of citizens insurance business is con- 
fined to three general lines, the practical problem being 
substantially the same in all cases. The three lines are 

1. Fire insurance, 

2. Life insurance, 

3. Accident insurance, 

and the practical problem is, Given the face of the policy 
and the rate to find the premium. 

Less common are such special forms as tornado, plate 
glass, and steam-boiler insurance, insurance against loss 
by theft, marine insurance, etc. The technical features of 
insurance are so constantly changing that it is inexpedient 
to enter into the subject with any detail. 

The premium is computed either as a certain per cent 
of the face of the policy, or, what is analogous to it, as a 
certain sum on each $100 of insurance. The latter is the 
usual form. Both this certain per cent and this certain 
sum go by the name rate of insurance. 

E.g., the rate for insuring the life of a man 30 yrs. old in a certain 
company, the policy to mature at death, is $22.85 annually on $1000, 
although it might be stated as 2.285%. 

The rate for insuring a business block against fire for 1 yr. (the 
usual time for insuring places of business) may be $1.10 on $100. 

The rate for insuring a house against fire for 3 yrs. (the usual time 
for insuring dwelling houses) may be $0.95 (for the 3 yrs.) on $100. 



178 HIGHER ARITHMETIC. 

Exercises. 1. What are the premiums for insuring business prop- 
erty against loss by fire for 1 yr. for the following amounts at the 
specified rates ? 

(a) $2000 at $0.90 per $100, contents for $5000 at $0.95 per $100. 

(6) $3500 " $1.10 " " $10,000 " $1.25 

(c) $8000 " $1.35 " " $50,000 ' $1.40 " 

(d) $7500 " $1.20 " " $35,000 " $1.30 " 

2. What are the premiums for insuring dwelling property against 
loss by fire for 3 yrs. for the following amounts at the specified rates ? 

(a) $1000 at $0.90 per $100, contents for $1500 at same rate. 
(6) $7000 " $1.10 " " $5000 " 

(c) $6500 " $0.95 " $4000 

(d) $4000 " $1.05 " " $3750 " 

3. What are the premiums for insuring manufacturing establish- 
ments against loss by fire for 1 yr. for the following amounts at the 
specified rates ? 

(a) $10,000 at $2.25 per $100. 
(6) $50,000 " $1.75 " 

(c) $25,000 " $1.95 

(d) $15,000 " $2.10 " 

4. What is the annual premium for insuring against loss by fire a 
business block for $8000 at $1.10 per $100, its ground and first floor 
contents for $10,000 at $1.20 per $100, its other contents for $8000 at 
$1.35 per $100, and two plate glass windows against damage from 
other causes than fire at $2.70 per window ? 

5. What is the annual premium for insuring a leaded glass window 
in a church for $1250 at 2% ? 

6. How much would be the annual premiums paid by a man 30 yrs. 
old for $5000 of life insurance at $22.85 per $1000 ? on the 10-pay- 
ment plan (of paying only ten times, the policy maturing at death) at 
$54.65 per $1000 ? on the 25-payment plan at $28.46 per $1000 ? on . 
the single-payment plan at $428.14 per $1000 ? 

7. As in Ex. 6, on a 10-year endowment policy (one in which ten 
payments are made, the policy then maturing, or maturing at death if 
before 10 yrs.) for $5000 at $106.75 per $1000 ? on a 25-year endow- 
ment policy for $5000 at $38.85 per $1000 ? 

8. What is the premium on a $1000 tornado insurance policy for 
1 yr. at 20 cts. per $100 ? for 3 yrs. at 50 cts. per $100 ? for 5 yrs. at 
80 cts. per $100 ? 



CHAPTER XXII. 
Miscellaneous Exercises. 



1. Multiply 1854.362 by 0.000087931, correct to 0.000001. 

2. Multiply 162.5473 by 8726.47231, correct to 0.0001. 

3. The distance of the moon from the earth is 59.97 times the 
earth's radius ; if this radius is 3962.824 mi., find the distance to the 
moon, correct to 1 mi. 

4. Divide 634.7538292 by 0.0657391, correct to 0.001. 

5. Divide 15.63214725 by 0.0057123, correct to 0.001. 

6. How many days, hours, minutes, and seconds in a year of 
365. 24226 da.? 

7. How often does the heart beat in a life of 75 yrs. of 365 da. 
each, supposing that the number of beats is 140 per min. during the 
first 3 yrs. of life, 120 for the next 3, 100 for the next 6, 90 for the 
next 10, 75 for the next 28, 70 for the next 20, and 80 for the last 5 ? 

8. Knowing that 

1,040,318,228,677 = 2,870,564 X 362,407 + 5,741,129, 
state the quotient and the remainder from dividing 1,040,318,228,677 
by 2,870,564 ; also by 362,407. 

9. Supposing a person can count one hundred in 30 sees. , and that 
after counting incessantly for 30 yrs. he dies, and his son goes on 
counting for 30 yrs. and then dies, and so on ; how many generations 
must elapse before one trillion is counted ? 

10. A person loses T ^ of his fortune and then T ^ of the remainder ; 
would the result have been the same if he had first lost ^ and then 

T ^ of the remainder ? Generalize for - and - 

11. How much is the fraction -^ 3 - increased or diminished when 5 
is added to each term ? 

12. In any year show that the same days of the month in March 
and November fall on the same day of the week. 



180 HIGHER ARITHMETIC. 

1H - 7, 



13. Reduce to simplest form 




14. Reduce the fraction g ' , 2 ^ 1X *> J^ J 7 to its simplest 
form. l(|-^i-AJ 

15. Simplify the expression T a of - 



i+=4 



16. Simplify the expression 



17. Divide 3 - f of T % by 21 J + & + 4 X 5. 

18. The first of a series of cog-wheels, working into one another 
in a straight line, has 7 n teeth ; the second has 6 w, the third 5 n, 
and the number in the fourth is to that in the third as 2 to 3. If the 
wheels are set in motion, how many revolutions must each make 
before they are simultaneously in their original positions ? 

19. Show that with a 1-ct. piece, two 2-ct. pieces, a 5-ct. piece, 
four dimes, a half-dollar, and nine silver dollars one can pay any sum 
less than or equal to $ 10. 

20. The moon revolves about the earth in 27 da. 7 hrs. 43 mins. 
11.5 sees. ; what is the average angle passed over in a day ? 

21. The length of an arc of 97 21' 47.2" is 23 in.; find, correct to 
0.1", the arc of the same circle 1 in. long. 

22. What fraction of the circumference is an arc of 27 17' 30" ? 
(Answer correct to 0.0001.) 

23. Prove that the sum of a common fraction and its reciprocal 
is greater than 2. Is there any exception ? 

24. Prove that if the same number is added to both terms of a 
fraction the new fraction is nearer unity than the old. 

25. Prove that, of three consecutive numbers, the difference 
between the squares of the first and third is four times the second. 

26. Prove that the difference between two numbers composed of 
the same digits, as 937 and 793, is a multiple of 9. 

27. Show that the integral part of a quotient is not changed by 
adding to the dividend a number less than the difference between the 
divisor and the remainder. 

28. Given the sum and the difference of two numbers, show how 
to find each. Prove your statement. 

29. Is the product of two square numbers always a square ? 

30. Prove that no number ending in 5 and not in 25 can be square. 



MISCELLANEOUS EXERCISES. 181 

31. Prove that a fraction whose terms are composed of the same 
number of digits is not altered in value by repeating the same number 
of times the figures of both terms. E.g., f| = ||f| = fHHI 

32. What is the common fraction which, reduced to a decimal 
fraction, equals 0.4275275 ? 

33. Prove that an integer cannot have for a square root a fractional 
number. 

34. Prove that any odd square number diminished by 1 is a mul- 
tiple of 8. 

35. State the test of divisibility of a number by 33. 

36. Show that if two numbers are prime to one another, any 
powers to which they may be raised are also prime to one another. 

37. Extract the 32d root of 429,497,296. 

38. Show that a factor of each of two numbers is also a factor of 
their greatest common divisor. 

39. If your school building is heated by a furnace, compare the 
area of a cross section of the cold-air pipe with the sum of the areas 
of cross sections of the hot-air pipes, and determine the ratio. 

40. What is the ratio of an arc of 321 22' to one of 37 21' 1" ? 

41. Divide the arc of 88 27' 33" into three parts proportional to 
the numbers 3.2, 5.6, 8.5. 

42. Divide the length of 28.75 in. into three parts proportional to 
the numbers f , f , T V 

43. How many pounds each of nickel and lead must be added to 
an alloy weighing 10 Ibs. and consisting of 11 parts (by weight) nickel, 
7 parts tin, and 5 parts lead, so that the new alloy shall consist of 19 
parts nickel, 41 parts tin, and 17 parts lead ? 

44. It takes a letter 43 da. to go from New York to Siam, a dis- 
tance of 12,990 mi., and 34 da. to go to Adelaide, Australia, a distance 
of 12,845 mi. What is the ratio of the average rate on the latter 
route to that on the former ? 

45. The area of Lake Superior is 32,000 sq. mi. and it drains an 
area of 85,000 sq. mi. ; the area of Lake Erie is 10,000 sq. mi. and its 
drainage is 39,680 sq. mi. Are the areas proportional to the drainage ? 
If not, what would be the drainage of Lake Superior to make them 
so? 

46. What force can a man weighing 165 Ibs. exert on a stone by 
pressing on a horizontal crowbar 6 ft. long, propped at a distance of 
5 in. from the point of contact with the stone, not considering the 
weight of the bar ? 



182 HIGHER ARITHMETIC. 

47. A uniform rod 2 ft. long weighs 1 Ib. ; what weight must be 
hung at one end in order that the rod may balance on a point 3 in. 
from that end ? 

48. Two men carry a weight of 20 Ibs. on a pole, one end being 
held by each ; the weight is 2 ft. from one end and 5 ft. from the 
other ; how many pounds does each support ? 

49. In a pair of nut crackers the nut is placed 1 in. from the hinge, 
and the hand is applied at a distance of 6 in. from the hinge ; if the 
nut requires a force of 22.5 Ibs. to break it, how much pressure must 
be exerted by the hand ? 

50. Three persons are associated in a common enterprise, the first 
having invested $4000, the second $7000, and the third $9000. At the 
end of a year their gains amount to $7340, out of which they pay the 
first $2000 for managing the business and divide the balance among 
the three in proportion to their investments. How much did each 
receive ? 

51. A country is 600 mi. long and 320 mi. wide ; find the dimen- 
sions of the paper on which a map of the country might be drawn, 
the scale being fa in. to the mile. 

52. The average number of deaths in the world each minute is 
estimated at 67, and the average number of births at 70 ; how many 
of each in a year of 365 da. ? 

53. By what fractional part of an inch should the highest mountain 
in Alaska, 19,500 ft., be represented on a globe 16 in. in diameter, the 
earth's radius being taken as 4000 mi. ? 

54. In a certain enterprise in which three persons are engaged, A 
puts in $3500 for 25 mo., B $2400 for 15 mo., C $4500 for 12 mo.; 
they gain $5000 ; what is the share of each ? 

55. A, B, and C rent a pasture for 6 mo. for $100 ; A puts in 25 
cattle for the whole time, B 30 for 4- mo., C 45 for 3 mo. ; find the 
rent paid by each. 

56. The streets of a certain city have an area of 8 km 2 . In a cer- 
tain storm the average depth of snow was 25 cm. Assuming 12 cm 3 
of snow to produce 1 cm 3 of water, find the volume of water produced 
by this snow, and the weight in metric tons. 

57. The wheels of a bicycle are 28 in. in diameter. The sprocket 
wheel connected with the pedals has 18 sprockets ; the other, 8. How 
many miles an hour does the rider make for one revolution of the 
pedals per sec.? If he travels 15 mi. per hr., how many revolutions 
of the pedals per min. ? 



MISCELLANEOUS EXERCISES. 183 

58. Milk gives about 20% in weight of cream, and cream gives 
about 30% in weight of butter. How many liters of milk will produce 
100 kg of butter, and how many kilograms of butter from 100 1 of 
milk ? The density of milk is 1.03. 

59. At a certain school rain fell one day to the depth of 36 mm. 
Calculate the volume of water which fell upon the school yard, a 
hectare in area ; also the weight of this volume of water ; also the 
respective weights of the oxygen and hydrogen contained, knowing 
that water is formed of eight parts in weight of oxygen to one of 
hydrogen. 

60. A cubic foot of water weighs 1000 oz., and in freezing expands 
^-Q of itself in length, breadth, and thickness ; find the weight of a 
cubic foot of ice, correct to 0.1. 

61. A liter of good milk weighs 1.030 kg. A milkman furnished 
4.5 1 of milk weighing 4.59 kg. Was there any water in it, and if 
so, how much ? 

62. How long will it take a man to walk around a square field 
whose area is 6f- acres, at the rate of a mile in 10|- mins. ? 

63. Having found the average number of inches of rainfall per 
yr. in your vicinity, determine the average number of gallons of 
water that fall upon the roof of your school building in one year. 

64. Of the world's supply of wool in a certain year, 2,456,733,600 
Ibs., Great Britain produced 147 Ibs. to the continent of Europe's 640 
and North America's 319 ; North America produced 6 Ibs. to Austral- 
asia's 11 ; Great Britain produced 15 Ibs. to the Cape of Good Hope's 
13 ; Australasia produced 22 Ibs. to the River Plate's 15 ; Australasia 
produces 577,500,000 Ibs ; how many pounds (correct to 100,000) were 
produced by all other countries together ? 

65. The value of the food-fishing industry in Alaska and Massa- 
chusetts together in a certain year was $8,149,987, Massachusetts 
exceeding Alaska by $3,547,877 ; what was the value in each ? 

66. A gas holder is to be constructed in the form of a circular 
cylinder, such that the radius of the circle shall be equal to the height ; 
find its dimensions to contain 100,000 cu. ft. of gas. 

67. If a terrestrial globe is constructed 36 in. in diameter, find the 
size on its surface of the United States, whose area is 3,500,000 sq. mi., 
the earth's diameter being 8000 mi. 

68. If 100 Ibs. of copper are drawn into 1 mi. of wire, find the 
diameter, copper being 8.9 times as heavy as water and 1 cu. ft. of 
water weighing 1000 oz. avoirdupois. 



184 HIGHER ARITHMETIC. 

69. An India rubber band 8 in. long, in. wide, T ^ in. thick, is 
stretched until it is 18 in. long and J in. wide ; what is then its thick- 
ness, assuming the volume to remain constant ? 

70. What is the amount of pressure exerted against one side of the 
upright gate of a canal, the gate being 24 ft. wide and submerged to 
the depth of 10 ft. ? 

71. A locomotive traveled for 32 sees, on a certain railroad at the 
rate of 112 mi. per hr.; how many revolutions were made in this 
time by the driving wheels, which were 78 in. in diameter ? 

72. If a man whose body has a surface of 15 sq. ft. dives in fresh 
water to the depth of 70 ft., what pressure does his body sustain ? 

73. Sea- water weighing 64.05 Ibs. per cu. ft., what is the pressure 
per sq. in. at the depth of 4655 fathoms of 6 ft. ? 

74. What length of paper yd. wide will be required to cover a 
wall 15 ft. 8 in. long by 11 ft. 3 in. high, no allowance being made 
for matching ? 

75. The diagonals of a quadrilateral field are 24 chains and 35 
chains in length respectively, and are perpendicular to one another ; 
how many acres in the field ? 

76. How many degrees in an arc equal in length to the radius of 
the circle ? 

77. A circle is inscribed in a square, the radius of the circle being 
8.5 in.; find the area between the sides of the square and the circum- 
ference of the circle. 

78. Find the difference between the area of a circle 15.4 in. in 
diameter and that of a regular inscribed hexagon. (The side of a 
regular inscribed hexagon equals the radius of the circle.) 

79. Three circles each 4 ft. in diameter touch one another ; find the 
area of the triangular figure enclosed by them. 

80. What is the length of the edge of the largest cube that can be 
cut out of a sphere 1 ft. in diameter ? 

81. What is the length of the edge of a cube whose surface is 
9 sq. ft. 54 sq. in. ? 

82. A bar of metal 9 in. wide, 2 in. thick, and 8 ft. long weighs 
1 Ib. per cu. in. ; find the length and thickness of another bar of the 
same metal, and of the same width and volume, if 2 in. cut off from 
the end weighs 27 Ibs. 

83. For a period of a week note carefully the number of minutes 
spent in the preparation of each of your various lessons. Represent 
the averages for the various subjects graphically. 



MISCELLANEOUS EXERCISES. 185 

84. Gunpowder being composed of sulphur, 75% niter, and the 
balance charcoal, how many pounds of each in 200 Ibs. of powder ? 

85. Three persons contribute $1000, $1200, $1780 respectively, and 
after trading 15 yrs. dissolve partnership ; the firm then being worth 
$18,000, what did each man receive ? 

86. A piece of wood which weighs 70 oz. in air has attached to it 
a piece of copper which weighs 36 oz. in air and 31.5 oz. in water ; 
the united mass weighs 11.7 oz. in water ; what is the specific gravity 
of the wood ? 

87. Find the weight of 10 mi. of steel wire 0.147 in. in diameter, 
the specific gravity being 7.872. 

88. Find the weight of a cast iron cylinder 8 ft. long, with a radius 
of 3.5 in., assuming the specific gravity to be 7.108. 

89. What is the weight of a circular plate of copper 11 in. in 
diameter and f in. thick, copper weighing 549 Ibs. per cu. ft.? 

90. What is the weight of a slate blackboard 19.5 ft. long, 3.5 ft. 
wide, and f- in. thick, the specific gravity of the slate being 2.848 ? 

91. If 31 cm 3 of gold weighs 599 g, find the specific gravity of gold. 

92. Find the time between 3 and 4 o'clock when the hour and 
minute hands of a watch are (1) together, (2) opposite, (3) at right 
angles to one another. 

93. A locomotive is going at the rate of 45 mi. per hr. ; how many 
revolutions does the drive-wheel, 22 ft. in circumference, make in a 
second ? 

94. How many seconds will a train 184 ft. long, traveling at the 
rate of 21 mi. per hr., take in passing another train 223 ft. long, 
going in the same direction at the rate of 16 mi. per hr.? how many 
seconds if they are going in opposite directions ? 

95. How many telegraph poles 58 ft. apart will a traveler by train 
going at the rate of 48 mi. per hr. pass in a minute ? 

96. A train leaves C for M at 9 A.M., traveling at a uniform rate 
of 15 mi. per hr. ; an express train leaves M for C at 10 A.M. at 
40 mi. per hr. ; at what time will they meet, and at what distance 
from C, the distance from C to M being 50 mi. ? 

97. A sledge party travels northward on an ice-floe at the rate of 
12 mi. per da. ; the floe is itself drifting eastward at the rate of 100 
rods per hr. ; at what rate is the sledge really moving ? 

98. A starts out on a bicycle at the rate of 8 mi. per hr. ; after 
he has gone 3 mi., B follows at the rate of 10 mi. per hr. ; after how 
many hours will B overtake A ? 



186 HIGHER ARITHMETIC. 

99. Out of a circle 18 in. in diameter there is cut a circle 13.5 in. 
in diameter ; what per cent of the original circle is left ? 

100. A laborer asks to have his time changed from 10 hrs. to 8 hrs. 
a day without decrease of daily pay ; by what per cent of his hourly 
wages does he ask them to be increased ? 

101. There was formerly in use a discount known as " true dis- 
count," which was the interest on the present worth of a given 
amount, that is, on such a sum as placed at interest for the given 
time should equal the given amount ; show that the bank discount 
equals the true discount plus the interest on the true discount. 

102. What is the present worth of $1356.80 due in 1 yr. 4 mo., the 
rate being 4|% ? 

103. What is the present worth and true discount of $1120 due in 
2 yrs. at 6% ? 

104. What is the present worth and true discount of $1000 due in 
11 mo. at 5% ? 

105. What is the present worth and true discount of $1430.40 due 
in 16 mo., the rate being 3% ? 

106. A dealer sells a machine for $80, taking a note to be paid 
without grace in 8 equal monthly payments without interest ; after 
two payments he takes the note to a bank and discounts it at 6% ; 
find the proceeds. 

107. A dealer sells a bicycle for $50, taking a note to be paid 
without grace in 10 equal monthly payments without interest ; after 
4 payments he discounts the note at 5% ; find the proceeds. 

108. A man sells a lot for $500, taking a note to be paid without 
grace in 10 equal monthly payments with interest at 5% ; after half of 
the payments have been made he discounts the note at 6% ; find the 
proceeds. 

109. A bookseller agrees to furnish a certain number of books for 
$66.30, after giving a discount of 15% upon the list prices. He him- 
self gets a discount of 25%. What is his gain ? 

110. A house cost $5000 and rents for $25 a month, with $25 to 
pay annually for repairs and $50 for taxes ; what is the difference in 
the income from this and from the same money invested in 6% stock 
at 96? 

111. A certain set of books in 10 volumes is offered for $66.60 
cash, or $69 payable $5 cash and $5 each succeeding month until the 
total amount of $69 has been paid. Which is the cheaper arrange- 
ment, money being worth 



MISCELLANEOUS EXERCISES. 187 

112. A man invests f of his money at 4% and the rest at 4|%. His 
annual income is $997.60. What is the ratio between the two parts 
of his income ? What are the two amounts invested ? What is the 
average interest upon his capital ? 

113. On a mortgage for $1700 dated May 28, 1900, there was paid 
Nov. 12, 1900, $80; Sept. 20, 1901, $314; Jan. 2, 1902, $50; Apr. 17, 
1902, $160 ; what was due Dec. 12, 1902, at 6% ? 

114. Which investment yields the better rate of income, one of 
$4200, yielding $168 semi-annually, or one of $7500, producing $712.50 
annually ? 

115. If an agent's commission is $290.40 when he sells $11,606 
worth of goods, how much would it be when he seljs $7416 worth ? 

116. What is the difference on a bill of $1750 between a discount 
of 40% and a discount of 30% 10% ? 

117. What per cent on the cost is gained by selling goods at the 
list price, they having been purchased at "a quarter off " ? 

118. How long will it take a sum of money to double itself at 6% 
simple interest ? at 6%, compounded semi-annually ? at 6%, com- 
pounded quarterly ? (Answer correct to 0.001.) 

119. The cost of maintaining the life-saving service of the United 
States for a certain year was $1,345,324 ; the value of the property 
saved was $9,145,000, and the value of the property involved was 
$10,647,000 ; these values were what per cent of the cost of maintain- 
ing the service ? 

120. The average number of hours which it takes for a letter to 
go from New York to London is 162.5 by one route and 176.7 by 
another ; what per cent is gained by taking the shorter passage ? 

121. The total valuation of farms in the United States in a certain 
year was $13,279,252,649, of the implements $494,247,467, and of 
the live stock $2,208,767,573; the value of farm products was 
$2,460,107,454. The value of the products was what per cent of the 
total valuation of farms, implements, and live stock ? 

122. After decreasing 13% from the acreage in a certain year, the 
number of acres in corn in the United States the following year was 
62,671,724 ; what was the acreage in the former year, correct to 1000 ? 

123. The United States produced 4,019,995 tons of steel in a cer- 
tain year, and that produced by other countries constituted 65.26% 
of the total production of the world ; required the total production. 

124. In a certain year the combined capital of all the fire insurance 
companies of the United States was $70,225,220 ; the total income for 



188 HIGHER ARITHMETIC. 

that year was $175,749,635, the amount paid for losses $89,212,971, 
for expenses and surplus $54,203,408, the balance going to dividends ; 
what was the average rate of dividend ? 

125. 59.9% of the total production of copper in the world comes 
from outside the United States ; how many tons does this represent, 
the production of the United States being 105,774 tons annually ? 

126. The dividends paid by the National Banks in a certain year 
were $45,333,270, the capital being $672,951,450; what was the 
average rate of dividend ? (Answer to 0.1%.) 

127. Find z, given (a) 5* = 20 ; (6) 100* = 2 ; (c) 8* = 100 ; 
(d) 4*+! = 50. 

128. Show that log (1 + 2 + 3) = log 1 + log 2 + log 3. Is this 
true f or 1 + 2 ? f or 1 + 2 + 3 + 4 ? 

129. Why is 10 the most practical base for a system of logarithms ? 
Why can 1 not be the base of a system ? 

130. What is the logarithm of 125 to the base 6 ? of 729 to the 
base 3 ? of 64 to the base 4 ? to the base 2 ? 

131. Show that if 3 is the base of the system of logarithms, 
log 81 + log 243 = log 19683, by finding each logarithm. 

132. We write numbers on a scale of 10 ; that is, we write up to 

10, then to 2 10, then to 3 10, 10-10 Show that we might 

write on a scale of 9 using one less digit ; or on a scale of 8 using two 
less digits, and so on to a scale of 2. 

133. How many characters are necessary for the scale of 10 ? of 
8 ? of 12 ? of 100 ? of n ? 

134. What characters are needed in the scale of 2 ? Write 15 
on the scale of 2. 

135. Write the numbers 12, 144, 145, 155, 1728, 1738 on the scale 
of 12. (The letters , e may be taken to stand for 10 and 11.) 

136. Add 1164 and 2345 on the scale of 7. 

137. From 3542 take 1164 on the scale of 7. 

138. Multiply 3542 by 4 on the scale of 7. 

139. Write 25 and its nth power, on the scale of 5. 

140. Write the numbers 15 and 21 on the scale of 2 ; multiply one 
by the other and hence show that if we used the scale of 2 it would 
not be necessary to learn the multiplication table. 

141. On the scale of 10 one-half is written 0.5 ; on the scale of 12 

it is written 0.6. Write the fractions , |, , $ as decimals; 

also write them on the scale of 12, and hence show that the scale of 
12 is adapted to computation better than the scale of 10. 



-APPENDIX. 



NOTE I, to p. 36. THE THEORY OF SQUARE BOOT 
CONTINUED. 

Trial divisor. The expression 2f is often called the 
trial divisor, 2f+n being called the complete divisor. It 
will be noticed in the example on p. 36 that the second 
trial divisor, 2/ 2 , equals the sum of the first complete 
divisor, 2/ x + n i> and n^ In other words, the new trial 
divisor can always be found by adding n to the last com- 
plete divisor. 

For 2/i + HI, added to m, equals 2 (/i + m). 

But /i + m= / 2 , .-. 2 (/i + m) = 2/ 2 . 

In the extraction of the square root of a long number 
like 299,066.7969, the ordinary abridged process may be 
still further shortened. In this example, it will be noticed 
that after the first three figures of the root have been found 
the next two can be found by merely dividing the remainder 
950.79 by the trial divisor. 

5 4 6. 8 7 0.87 

29<90'66.79'69 1092 ) 950.79 

104 4 90 77.19 

1086 7466 .75 

1092.8 950.79 
1093.67 76.55 69 



That this is true in general in the case of a perfect square 
will now be proved. 



190 HIGHER ARITHMETIC. 

Abridgment theorem. After n -f 1 figures of the square 
root of a perfect second power (or square) have been 
found, the next n figures can be found by dividing the 
next remainder by the next trial divisor. 

1. Let Vs =/ + , where s is a perfect second power, / contains 
n + 1 figures already found, and x contains n figures to be found. 

2. v the n + 1 figures of / are followed by the n figures of x, f has 
the value of a number of 2 n + 1 figures. 

3. From 1, s = / 2 + 2/x + x*, or ^^ = x + ^ That is, if 

*/ ^/ 

the remainder s / 2 is divided by the trial divisor 2/, the quotient 

x 2 
is x plus the fraction 

x 2 

4. .-. if it is shown that is a proper fraction, the integral part of 

x 2 

x + is x, the remaining part of the root, and the theorem is proved. 
*/ 

5. v x contains n figures, .-. x < 10", and x 2 < 10 2 . 

6. v / has the value of a number of 2 n + 1 figures, .-. / < 10 2n , 
and 2/ < 2 10 2w . 

x 2 10 2n 

7. .- gy < 2 . 1()2M or i, a proper fraction. 

Exercise. Show that the abridgment theorem holds only for a 
perfect second power as stated, by considering the cases of 152,399,000, 
and 152,399,025, the latter being a perfect second power. 



NOTE II, to p. 40. THE THEORY OF CUBE BOOT 
CONTINUED. 

Trial divisor. The expression 3/ 2 is often called the 
trial divisor, 3fn -\- n 2 being called the correction, and 
3/ 2 + 3/w + n 2 the complete divisor. It will be noticed 
in the example on p. 39 that the second trial divisor, 
3/ 2 (or 780,300), equals the sum of the first complete 
divisor (3/ 2 + 3fn + n 2 , or 765,100) and the correction 
(Sfn + n 2 , or 15,100) and the square of n (n 2 , or 100). 
This is always true in cube root. 



APPENDIX. 191 

For / t =/i H- m. 

.'. 3/ 2 2 = 3 (/i + m) = 3A 2 + 6/mi + 3 m 

= 3/i 2 + 3/ini + wi 2 (the preceding complete divisor) 
+ 3/iWi + wi 2 (the correction) 

+ ni 2 (the square of n). 

And the same reasoning holds for all trial divisors in cube root. 
This is much shorter than the operation of squaring / and multi- 
plying by 3 each time. 

Abridgment theorem. After n + 2 figures of the cube 
root of a perfect third power have been found, the remain- 
ing n figures can be found by dividing the next remainder 
by the next trial divisor. 

1. Let vT = /+x, where t is a perfect third power, / contains 
n + 2 figures already found, and x contains n figures to be found. 

2. v the n + 2 figures of / are followed by the n figures of x, / has 
the value of a number of 2 n + 2 figures. 

3. From 1, t =/ + 3/ 2 x + 3/x 2 + x 3 , 

... ^^ = x + + -j~ ' That is ' if the remainder t ~f* is 
divided by the trial divisor 3/ 2 , the quotient is x plus the fractions 

x 2 . x* 
7 and 

x 2 x s 

4. .-. if it is shown that + -^-^ equals a proper fraction, the 

integral part of the quotient is x and the theorem is proved. 

5. v x contains n figures, .-. x < 10 M , x 2 < 10 2 , and x 3 < 10*. 

6. v / has the value of a number of 2 n + 2 figures, 
... /< iQ2+i, /2 < lQ4+2, and 3/ 2 < 3 10 4 + 2 . 

x 2 
7- 



x 2 10 2 1 


" / K 
x 3 


,2+l' ' A 10 ' d 

10 3 1 


3/ 2 '^ 3 

. x 2 x 


. !Q4n+2 ' 01 3 lQn+2 
3 1 1 

<- 4- 


'/ ' 3, 


r 2 ^io 3-io+ 2 ' 



8. .'. j + ~ < + 37-2 , a proper fraction. 

Exercise. Show that with the first three figures of the square root 
of 14,696,712,600,000,000, obtained by the ordinary process, the next 
two cannot be correctly found by division. (Similar conditions evi- 
dently exist in the theory of cube root.) 



192 HIGHER ARITHMETIC. 

Three cube roots. (Omit if the class has not studied 
quadratic equations and imaginaries.) Just as 4 has two 
square roots, + 2 and 2, so 8 has three cube roots, 2, 
1 + V 3, and 1 V 3, as may easily be proved 
by cubing. So in general, every number has three and 
only three cube roots. 

1. For let x 3 n ; then if the value of x is found the cube root of 
n is known. 

2. From 1, x 3 n = 0. 

3. .-. (x Vn)(z 2 + x Vw + \^ = 0, by factoring (2). 

4. This equation is satisfied if either 



x \ = 0, or x 2 + x tt + Vw 2 = 0. 

5. If x Vn = 0, then x = Vn, the ordinary arithmetical cube 
root. 

6. If x 2 + x Vn + Vn 2 = 0, then, by solving the quadratic equation, 



7. That is, the three cube roots of n are 



Exercises on the three cube roots. 1. What are the three 
cube roots of 1 ? Verify your answer by cubing. 

2. Show that each of the imaginary cube roots of 1 is the square of 
the other. 

3. Show that the sum of the three cube roots of any number is zero. 

4. Show that the product of the three cube roots of any number is 
that number. 

5. Show that any number has four fourth roots. 

NOTE III, to p. 146. THE THEORY OF COMPOUND 

INTEREST CONTINUED. 

The theory gf compound interest affords an application 
of logarithms for those who have studied Chap. XI. The 
work on p. 193, excepting steps 1-4 and Ex. 1, should be 
omitted by others. 



APPENDIX. 193 

E.g., by logarithms the computation on p. 145 becomes simple : 

log 1.02 = 0.0086 

6 log 1.02 = 0.0516 

log 150 = 2.1761 

2.2277 = log 168.9, 
as near as the result can be obtained by the small table on p. 114. 

The principal formulae are deduced as follows : 

1. Let p the number of dollars of principal, r the rate 
at which the interest is to be compounded annually, t the 
number of years, and a 1} a 2 , a s , ..... a t the amounts for 1, 2, 
3, ..... t years. 

2. Then, ai = p-{-rp = (l + r)p, 

a, = (1 + r)p + r (1 + r)p = (1 + rfp, 
a 8 = (1 + r) 8 p, and in general 

3. a t =(l + ryp. 

'- 



5. From 3, log a, = t log (1 + r) + log p. 
. log , logy 
log (! + ) 

7. And from 5, r = antilog loga '~ logy - 1. 

If the interest is compounded semi-annually for t yrs. at r% a 
year, the amount is evidently the same as if the interest were com- 

pounded annually for 2 1 yrs. at -% a year. 

u 

Exercises. 1. What is the amount of each of the following ? 

FACE. TIME. RATE. COMPOUNDED. 

(a) $250 3 yrs. 2 mo. 10 da. 4% semi-annually. 

(6) $75 1 yr. 4 " 3 " 4% quarterly. 

2. Find the principal, given the 

AMOUNT. TIME. RATE. COMPOUNDED. 

(a) $384.03 5 yrs. 5% semi-annually. 

(6) $162.36 2 " 4% 

3. In how many years will $200 amount to $310.26 at 5%, com- 
pounded annually ? 



Plane and Solid Geometry, 

BY 

WOOSTER WOODRUFF BEMAN, 

Professor of Mathematics in the University of Michigan, 

AND 

DAVID EUGENE SMITH, 

Professor of Mathematics in the Michigan State Normal School. 



12mo, Half leather, ix + 320 pages, For introduction, $1,25, 



While not differing radically from the standard American high school 
geometry in amount and order of material, this work aims to introduce 
and employ such of the elementary notions of modern geometry as will 
be helpful to the beginner. 

Among these are the principles of symmetry, reciprocity or duality, 
continuity, and similarity. 

The authors have striven to find a happy mean between books that 
leave nothing for the student to do and those that throw him entirely 
upon his own resources. At first the proofs are given with all detail ; 
soon the references are given only by number; and finally much of the 
demonstration must be wrought out by the student himself. 

The exercises have been carefully graded and have already been put 
to a severe test in actual practice. Effort has been made to guide in 
the solution of these exercises by a systematic presentation of the best 
methods of attacking " original " theorems and problems. 

The leading text-books on geometry in English and other languages 
have been examined and their best features, so far as thought feasible, 
incorporated. 

These merits will be found enhanced, it is believed, by the beauty 
and accuracy of the figures, and the excellence of the typographical 
make-up. 

Wentworth's Geometry has never been so widely used as now, nor has 
it seemed to give satisfaction more uniformly, but the publishers believe 
that this new book will be cordially welcomed and that many teachers 
will find it admirably suited to the needs of their classes. 



GINN & COMPANY, PUBLISHERS, 

BOSTON. NEW YORK. CHICAGO. 



MATHEMATICAL TEXT-BOOKS. 

For Higher Grades. 

Anderegg and Roe: Trigonometry $0.75 

Andrews: Composite Geometrical Figures 50 

Baker: Elements of Solid Geometry 80 

Eeman and Smith: Plane and Solid Geometry 1.25 

Byerly: Differential Calculus, $2.00 ; Integral Calculus 2.00 

Fourier's Series 3.00 

Problems in Differential Calculus 75 

Carhart: Field-Book, $2.50 ; Plane Surveying 1.80 

Comstock: Method of Least Squares 1.00 

Faunce : Descriptive Geometry 1.25 

Hall: Mensuration 50 

Halsted: Metrical Geometry 1.00 

Hanus : Determinants 1.80 

Hardy: Quaternions, $2.00 ; Analytic Geometry 1.50 

Differential and Integral Calculus 1.50 

Hill : Geometry for Beginners, $1.00 ; Lessons in Geometry.. .70 

Hyde: Directional Calculus 2.00 

Macf arlane : Elementary Mathematical Tables 75 

Osborne : Differential Equations 50 

Peirce (B. 0.): Newtonian Potential Function 1.50 

Peirce (J. M.) : Elements of Logarithms, .50 ; Mathematical Tables.. .40 

Kunkle: Plane Analytic Geometry 2.00 

Smith: Coordinate Geometry 2.00 

Taylor: Elements of the Calculus 1.80 

Tibbets : College Requirements in Algebra 50 

Wentworth: High School Arithmetic 1.00 

School Algebra, $1.12; Higher Algebra 1.40 

College Algebra 1.50 

Elements of Algebra, $1.12: Complete Algebra 1.40 

New Plane Geometry 75 

New Plane and Solid Geometry 1.25 

Plane and Solid Geometry and Plane Trigonometry.. 1.40 

Analytic Geometry 1.25 

Geometrical Exercises 10 

Syllabus of Geometry 25 

New Plane Trigonometry 40 

New Plane Trigonometry and Tables 90 

New Plane and Spherical Trigonometry 85 

New Plane and Spherical Trig, with Tables 1.20 

New Plane Trig, and Surveying with Tables 1.20 

New Plane and Spher. Trig., Surv., with Tables 1.35 

New Plane and Spher. Trig., Surv., and Navigation.. 1.20 

Wentworth and Hill: Exercises in Algebra, .70; Answers 25 

Exercises in Geometry, .70; Examination Manual 50 

Five-place Log. and Trig. Tables (7 Tables) 50 

Five-place Log. and Trig. Tables (Complete Edition) 1.00 

Wentworth, McLellan, and Glashan : Algebraic Analysis 1.50 

Wheeler : Plane and Spherical Trigonometry and Tables 1 .00 



Descriptive Circulars sent, postpaid, on application. 
The above list is not complete. 



OINN & COMPANY, Publishers, 

Boston. New York. Chicago. Atlanta. Dallas. 



Algebra Reviews 



EDWARD RUTLEDGE ROBBINS, 

Master in Mathematics and Physics in the La-wrenceville School, 
Laiurenceville, N.J. 



12mo, Paper. 44 pages, For introduction, 25 cents. 



THIS little book is intended to be used only during review 
and in place of the regular text-book in elementary algebra. 
At this period the teacher is often taxed in the selection of 
the best examples, and the pupil confused and tired by the 
long and scattered review lessons. 

A few illustrative solutions in each of the several topics of 
algebra occupy the left-hand page, while the right-hand page 
is filled with particular examples on the same subject. Thus 
the book is believed to present the essentials of algebra 
briefly and completely. 

There has been added a useful list of eleven representative 
examinations, the examples of which are chiefly taken from 
recent college entrance papers. All theory, definitions, and 
the answers to many of the simpler problems have been 
omitted. 

S. B. Tinsley, Teacher of Mathematics, Male High School, Louisville, 
Ky. : It is an excellent little book and should meet with the approval 
of all teachers of algebra. 

G. F. Cook, Professor of Mathematics, South West Kansas College, 
Winfield, Kan. : The problems are so well arranged and so carefully 
selected that every page is worth an ordinary chapter. 

W. A. Obenchain, President of Ogden College, Bowling Green, Ky. : 
It is an excellent little book. I shall certainly use it in my classes next 
year. 



GINN & COMPANY, PUBLISHERS, 

Boston. New York. Chicago. Atlanta. Dallas. 



A PRACTICAL ARITHMETIC 

By GEORGE A. WENTWORTH, 

Author of the Went-worth Series of Mathematics. 



12mo, Half leather, 344 pages, Illustrated, For introduction, 65 cents, 



THE name Practical Arithmetic indicates the character 
of the book. It is meant for everyday conditions. The 
language is simple, avoiding technical expressions so far as 
possible. Statements are clear and direct. Definitions are 
sharp. Explanations explain. Rules are crisp and precise. 
The examples are new, progressive, and practical. They 
are especially adapted to securing mental discipline, and a 
practical mastery of the common problems of everyday life. 

The problems involve no puzzles or unnecessary tedious- 
ness of figuring. They illustrate and apply the rules and 
principles, without requiring more power of thought than the 
pupils may safely be expected to possess. They are based 
upon the idea of correlation of studies, involving the facts 
of the other school studies in such a way as to fix and illus- 
trate them. 

This is a book for all schools, easily taught, and sure to 
be mastered by the diligent pupil. It is designed to fix 
principles and rules so clearly in the mind that they will 
never be forgotten. 

The whole trend of the book is calculated to make pupils 
think, to cultivate the reasoning powers, to enlist all their 
attention, and to put knowledge in a form for practical use. 

In brief, we have every reason to believe that Went- 
worth' s Practical Arithmetic is the best arithmetic of its 
grade to give a real mastery of the subject. 



GINN & COMPANY, PUBLISHERS, 

BOSTON. NEW YORK. CHICAGO. ATLANTA. DALLAS. 



Wentworth's Mathematics 



BY 



GEORGE A, WENTWORTH, A.M. 



ARITHMETICS 

They produce practical 
arithmeticians . 

Elementary Arithmetic ... $ .30 

Practical Arithmetic 65 

Mental Arithmetic 30 

Primary Arithmetic 30 

Grammar School Arithmetic . . .65 
High School Arithmetic . . . i.oo 
Wentworth & Hill's Exercises 

in Arithmetic (in one vol.) . .80 
Wentworth & Reed's First Steps 

in Number. 

Teacher's Edition, complete . . .90 
Pupil's Edition 30 



ALGEBRAS 

Each step makes the 
next easy. 

First Steps in Algebra ... $ .60 

School Algebra 1.12 

Higher Algebra 1.40 

College Algebra 1.50 

Elements of Algebra 1.12 

Shorter Course i.oo 

Complete 1.40 

Wentworth & Hill's Exercises 

in Algebra (in one vol.) . . .70 



THE SERIES OF THE GOLDEN MEAN 



GEOMETRIES 

The eye helps the mind to grasp 
each link of the demonstration. 

New Plane Geometry. ... $ .75 
New Plane and Solid Geometry. 1.25 
P. & S. Geometry and Plane 

Trig 1.40 

Analytic Geometry 1.25 

Geometrical Exercises 10 

Syllabus of Geometry 25 

Wentworth & Hill's Examina- 
tion Manual 50 

Wentworth & Hill's Exercises . .70 



TRIGONOMETRIES, ETC. 

Directness of method secures 
economy of mental energy. 

New Plane Trigonometry . . $ .40 
New Plane Trigonometry and 

Tables go 

New Plane and Spherical Trig. . .85 
New Plane and Spherical Trig. 

with Tables 1.20 

New Plane Trig, and Surv. with 

Tables 1.20 

Tables 50 or i.oo 

New P. and S. Trig., Surv., with 

Tables 1.3$ 

New P. and S. Trig., Surv., and 

Nav 1.20 

The old editions are still issued. 



GINN & COMPANY, Publishers, 

Boston. New York. Chicago. Atlanta. Dallas. 



BOOKS USEFUL TO STUDENTS OF 

ENGLISH LITERATURE. 



Arnold's English Literature. 558 pages. Price, $1.50. 
Baker's Plot-Book of Some Elizabethan Plays. In press. 
Baldwin's Inflection and Syntax of Malory's Morte d'Arthur. 
Browne's Shakspere's Versification. 34 pages. Price, 25 cts. 
Corson's Primer of English Verse. 232 pages. Price, $1.00. 
Emery's Notes on English Literature. 152 pages. Price, $1.00. 
Garnett's Selections in English Prose from Elizabeth to Victoria. 701 

pages. Price, $1.50. 

Gayley's Classic Myths in English Literature. 540 pages. Price, $1.50. 
Gayley's Introduction to Study of Literary Criticism. In press. 
Gummere's Handbook of Poetics. 250 pages. Price, $1.00. 
Hudson's Life, Art, and Characters of Shakespeare. 2 vols. 1003 pages. 

Price, $4.00. 

Hudson's Classical English Reader. 467 pages. Price, $1.00. 
Hudson's Text-Book of Prose. 648 pages. Price, $1.25. 
Hudson's Text-Book of Poetry. 704 pages. Price, $1.25. 
Hudson's Essays on English Studies in Shakespeare, etc. 118 pages. 

Price, 25 cts. 

Lee's Graphic Chart of English Literature. 25 cts. 
Minto's Manual of English Prose Literature. 566 pages. Price, $1.50. 
Minto's Characteristics of the English Poets. (From Chaucer to Shirley.) 

382 pages. Price, $1.50. 
Montgomery's Heroic Ballads. Poems of War and Patriotism. Edited 

with Notes by D. H. Montgomery. 3^9 pages. Boards, 40 cts.; 

Cloth, 50 cts. 
Phelps's Beginnings of the English Romantic Movement. 192 pages. 

Price, $1.00. 

Sherman's Analytics of Literature. 468 pages. Price, $1.25. 
Smith's Synopsis of English and American Literature. 125 pages. Price, 

80 cts. 

Thayer's Best Elizabethan Plays. 611 pages. Price, 11.25. 
Thorn's Shakespeare and Chaucer Examinations. 346 pages. Price, $1.00. 
White's Philosophy of American Literature. 66 pages. Price, 30 cts. 
Winchester's Five Short Courses of Reading in English Literature. 

99 pages. Price, 40 cents. 
Wylie's Studies in the Evolution of English Criticism. 212 pages. 

Price, $1.00. 

Descriptive Circulars of these books sent postpaid to any address. 



GINN & COMPANY, Publishers, Boston, New York, and Chicago. 



THE CLASSIC MYTHS 

IN 

ENGLISH LITERATURE. 

BY CHARLES MILLS GAYLEY, 

Professor of the English Language and Literature in the University of California 
and formerly Assistant-Professor of Latin in the University of Michigan. 



i2mo. Half leather. 540 pages. For introduction, $1.50. 
New Edition with 16 full-page illustrations. 

This work, based chiefly on Bulfinch's " Age of Fable " 
(1855), has here been adapted to school use and in large 
part rewritten. It is recommended both as the best manual 
of mythology and as indispensable to the student of our 
literature. 

Special features of this edition are : 

1. An introduction on the indebtedness of English poetry to the 
literature of fable ; and on methods of teaching mythology. 

2. An elementary account of myth-making and of the principal 
poets of mythology, and of the beginnings of the world, of gods and 
of men among the Greeks. 

3. A thorough revision and systematization of Bulfinch's Stories of 
Gods and Heroes : with additional stories, and with selections from 
English poems based upon the myths. 

4. Illustrative cuts from Baumeister, Roscher, and other standard 
authorities on mythology. 

5. The requisite maps. 

6. Certain necessary modifications in Bulfinch's treatment of the 
mythology of nations other than the Greeks and Romans. 

7. Notes, following the text (as in the school editions of Latin and 
Greek authors), containing an historical and interpretative commentary 
upon certain myths, supplementary poetical citations, a list of the bette* 
known allusions to mythological fiction, references to works of art^ 
and hints to teachers and students. 



GINN & COMPANY, Publishers, 

Boston, New York, and Chicago. 



WHAT IS SAID OF 



Spenser's Britomart 



Edward Dowden, Professor of English Literature in the University 
of Dublin, Ireland. 

It was a happy thought to bring together the beautiful story of 
Britomart, and to make many persons who know only Una, know 
Spenser's warrior heroine. I think all lovers of Spenser's poetry 
will say you have done a good deed. The notes aid a reader 
without interrupting his pleasure and the introduction says well 
all that it is needful to say. 

George L. Kittredge, Professor of English in Harvard University. 

I am greatly pleased with it. The editor's leading idea was 
decidedly a happy one ; and she has followed it with uncommon 
judgment and good taste. The introduction is precisely what was 
needed, and the notes are both simple and sufficient. The book 
should prove very useful to beginners in Spenser, and even those 
who are not beginners will find it attractive. 

J. Russell Hayes, Assistant Professor of English in Swarthmore 

College, Swarthmore, Pa, 

I am glad to see that you are doing some good work for the 
spreading of the study of Spenser. His great masterpiece has 
been too long neglected, and works like this of Miss Litchfield's 
will help to call to Spenser some of the attention which the schools 
devote, too exclusively, I think, to his greater contemporary. 

Arthur R. Marsh, Assistant Professor of Comparative Philology in 
Harvard University. 

I have examined the book with the greatest pleasure. It seems 
to me that the conception is most happy and the execution very 
judicious. I rarely get hold of a book of this kind that I can so 
unreservedly praise. I hope its excellence may lead to its being 
widely used in schools and colleges, and that it may serve to 
awaken a more general interest in one of the noblest and most 
delightful, though one of the least read, of English poets. 



QINN & COMPANY, 

BOSTON. NEW YORK. CHICAGO. ATLANTA. DALLAS. 



English Composition and Rhetoric 

Text-books and works of reference for 
high schools, academies, and colleges. 



Lessons in English. Adapted to the study of American Classics. A 
text-book for high schools and academies. By SARA E. H. LOCK- 
WOOD, formerly Teacher of English in the High School, New Haven 
Conn. Cloth. 403 pages. For introduction, $1.12. 

A Practical Course in English Composition. By ALPHONSO G. NEW- 
COMER, Assistant Professor of English in Leland Stanford Junior 
University. Cloth. 249 pages. For introduction, 80 cents. 

A Method of English Composition. By T. WHITING BANCROFT, late 
Professor of Rhetoric and English Literature in Brown University. 
Cloth. 101 pages. For introduction, 50 cents. 

The Practical Elements of Rhetoric. By JOHN F. GENUNG, Professor 
of Rhetoric in Amherst College. Cloth. 483 pages. For intro- 
duction, $1.25. 

A Handbook of Rhetorical Analysis. Studies in style and invention, 
designed to accompany the author's Practical Elements of Rhetoric. 
By JOHN F. GENUNG. Cloth. 306 pages. Introduction and teachers' 
price, $1.12. 

Outlines of Rhetoric. Embodied in rules, illustrative examples, and a 
progressive course of prose composition. By JOHN F. GENUNG. 
Cloth. 331 pages. For introduction, $1.00. 

The Principles of Argumentation. By GEORGE P. BAKER, Assistant 
Professor of English in Harvard University. Cloth. 414 pages. For 
introduction, $1.12. 

The Forms of Discourse. With an introductory chapter on style. By 
WILLIAM B. CAIRNS, Instructor in Rhetoric in the University of 
Wisconsin. Cloth. 356 pages. For introduction, $1.15. 

Outlines of the Art of Expression. By J. H. GILMORE, Professor of 
Logic, Rhetoric, and English in the University of Rochester, N.Y. 
Cloth. 117 pages. For introduction, 60 cents. 

The Rhetoric Tablet. By F. N. SCOTT, Assistant Professor of Rhetoric, 
University of Michigan, and J. V. DENNEY, Associate Professor of 
Rhetoric, Ohio State University. No. i, white paper (ruled). No. 2, 
tinted paper (ruled). Sixty sheets in each. For introduction, 1 5 cents. 

Public Speaking and Debate. A manual for advocates and agitators. 
By GEORGE JACOB HOLYOAKE. Cloth. 266 pages. For intro- 
duction, $1.00. 



QlNN & COMPANY, Publishers, 

Boston. New York. Chicago. Atlanta. Dallas. 



FULL OF LIFE AND HUMAN INTEREST 



H ISTORIBS 

FOR HIGH SCHOOLS AND COLLEGES, 

BY PHILIP VAN NESS MYERS, 

Professor of History and Political Economy in the University of Cincinnati, Ohio, 

AND 

WILLIAM F. ALLEN, 

Late Professor of History in the University of Wisconsin. 



Myers's General History. Half morocco. Illustrated. 759 pages. 

For introduction, $1.50. 
Myers's History of Greece. Cloth. Illustrated. 577 pages. For 

introduction, $1.25. 
Myers's Eastern Nations and Greece. (Part I. of Myers's and of Myers 

and Allen's Ancient History.) Cloth. 369 pages. For introduc- 
tion, $1.00. 
Myers and Allen's Ancient History. (Part I. is Myers's Eastern 

Nations and Greece. Part II. is Allen's Short History of the 

Roman People.) Half morocco. 763 pages. Illustrated. For 

introduction, $1.50. 
Myers's Ancient History. (Part I. is Myers's Eastern Nations and 

Greece. Part II. is Myers's Rome.) Half morocco. 617 pages. 

Illustrated. For introduction, $1.50. 
Myers's History of Rome. (Part II. of Myers's Ancient History.) 

Cloth. 230 pages. For introduction, $1.00. 
Allen's Short History of the Roman People. (Part II. of Myers and 

Allen's Ancient History.) Cloth. 370 pages. For introduction, 

$1.00. 
Myers's Outlines of Mediaeval and Modern History. Half morocco. 

740 pages. For introduction, $1.50. 

A philosophical conception of history and a broad view of its devel- 
opments, accurate historical scholarship and liberal human sympathies 
are the fundamental characteristics of these remarkable histories. The 
hand of a master is shown in numberless touches that illuminate the 
narrative and both stimulate and satisfy the student's curiosity. 

Schoolroom availability has been most carefully studied, and typo- 
graphical distinctness and beauty, maps, tables and other accessories 
have received their full share of attention. 



GIJNN & COMPANY, PUBLISHERS. 



UNIVERSITY OF CALIFORNIA LIBRARY 
BERKELEY 

Return to desk from which borrowed. 
This book is DUE on the last date stamped below. 



LD 21-100m-9,'48(B399sl6)476 



YB 17350 



911349 



THE UNIVERSITY OF CALIFORNIA LIBRARY 



